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FHE USES AND TRIUMPHS
MATHEMATICS
V.E. JOHNSON, B.A.
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THE USES AND TRIUMPHS
OF MATHEMATICS.
THE USES AND TRIUMPHS
OF MATHEMATICS
ITS BEAUTIES AND ATTRACTIONS
POPULARLY TREATED IN
THE LANGUAGE OF EVERYDAY LIFE
BY
V. E. JOHNSON, B.A.
MAGDALENE COLLEGE, CAMBRIDGE
' For by known principles you understand the most difficult subjects much
more easily.' — Cicero.
Without Mathematics, expressed or implied, our knowledge of Physics would
be friable in the extreme.' — Tyndall.
LONDON
GRIFFITH FARRAN OKEDEN & WELSH
(SUCCESSORS TO NEWBEKY AND HARRIS)
AND SYDNEY.
The Rights of Translation and of Reproduction are reserved.
LECTRUN1C VERSION
AVAILABLE
DEDICA TION.
To that long series of 'Illustrious Men who
by their labours and discoveries have ren
dered the powers of Nature the servants of
man, in contradiction to EMPIRICISM, which
subjects man to their service, this little Book
is respectfully dedicated by
THE AUTHOR.
CONT/LNTS.
CHAPTER I.
PAGE
INTRODUCTION, ...... I
CHAPTER II.
THE USES OF MATHEMATICS, . . . 2O
CHAPTER III.
THE TRIUMPHS OF MATHEMATICS, . • • 39
CHAPTER IV.
THE LIMITS OF MATHEMATICS, . . .63
CHAPTER V.
THE BEAUTY OF MATHEMATICS, ... 73
CHAPTER VI.
THE ATTRACTIONS OF MATHEMATICS, . . 87
CHAPTER VII.
THE POETRY OF MATHEMATICS, . . . 97
viii Contents.
CHAPTER VIII.
PAGE
METAPHYSICAL OR SPIRITUALISTIC MATHEMATICS, . 113
CHAPTER IX.
CONCLUSION, . . . . . .131
APPENDIX.
'THE SQUARING OF THE CIRCLK,' . . 139
PREFACE.
' ETHEL/ I remember once hearing a lady
say to her daughter, ' did Mr Antony say
he intended to bring a friend with him this
evening?' ' Yes, mamma, — a Mr Bertrand,
a mathematician.' 'Oh dear!' replied her
mother, ' what a wet blanket he will be !
I hope he will not be trying to " Square
the Circle," don't they call it, or some
thing of that sort ? '
Is not the opinion which this lady held
with regard to Mathematicians and Ma
thematics the one very often entertained,
namely, that Mathematics renders a man
x Preface.
unsociable, unpoetical, calculating, and with
out faith in anything which will not ad
mit of a rigid demonstration ; and that the
Science of Mathematics is a dreadful sub
ject, — a compound of fearful words and
symbols which only the initiated are able
to appreciate or perceive any use in ?
I shall endeavour to show, in a non-
mathematical manner, in the following pages,
that this Science can be put to every
possible kind of practical use, and also
to refute the statement that the Science
of Mathematics is a dreadfully dry subject,
without any beauty or poetry in it, — simply
a conglomeration of straight lines and
circles, a's and b's, x's and y's, sines and
cosines, etc. etc., but that it is one of the
most profound and the most fascinating,
and, in some respects, the most beautiful,
of all the Arts and Sciences.
Preface. xi
The object of this Essay is an attempt
to create a desire for the study of Mathe
matics, by showing its intimate and im
portant connection with so many branches
of Science (man's greatest helpmate), by
stating and explaining a few of its uses
and triumphs, and by attempting to prove
that it is a subject possessed of a beauty
and attraction entirely its own.
Much has been done at various times to
popularise the subject of Mathematics, but
much more remains to be done before this
Science will be divested of that perhaps
quite natural aversion which a non-mathe
matical person always experiences on open
ing any book treating on any of its various
branches. Some knowledge of Mathema
tics (or what is commonly called such)
is now required in every public examina
tion, and therefore the subject is too often
xii Preface.
taught and learned for these examinations,
and not on account of those uses, etc.
(excepting arithmetic), which the subject
possesses in itself, and as an aid to Science.1
The majority of students, having acquired
the necessary degree of skilfulness in the
manipulation of certain symbols, or a
mnemonic acquaintance with certain geo
metrical demonstrations, pass the examina
tion, and then drop the subject at once,
it having been to them only a means to
an end, — an abominably dry subject to be
avoided as soon as possible.
And this is not surprising, considering
the uninteresting way in which this sub
ject is usually presented to the student,
ignorant alike as he is of its wonderful and
1 ' The distinctive feature of mathematical instruction in
England is, that an appeal is there made rather to the
memory than to the intelligence of the pupil.' — Messrs
Demogeot and Montucci — Report on English Education.
Preface. xiii
varied uses ; of the attractions which this
subject possesses on account of its intimate
connection with so many departments of
Science and Art ; of its historical interest ;
and of what a vast though indirect influ
ence this Science has had on the progress
of the human race.
The contents of this little book is also
an attempt to meet in some measure this
want, being intended mostly for the non-
mathematical ; no mathematical knowledge
is required for its perusal ; and I have
endeavoured, as far as I have been able,
to make it as interesting as possible.
For much matter and many ideas I am
indebted to 'The Orbs of Heaven' (a book,
though antiquated, well worthy of a place
in the library of everyone), the writings of
R. W. Emerson, Professor Tyndall, Justus
Liebeg, Dr Whewell, Lord Bacon, R. A.
xiv Preface.
Proctor, C. Flammarion, and the articles
on Mathematics and Geometry in the
' National Encyclopaedia,' and various other
writers, to whom reference will be made
in due course. Whatever well expressed
thought or idea I have found (no matter
where) illustrating my subject, I have taken,
and, when possible, I have acknowledged it.
I have created but a small portion of the
materials of this work. The only claim to
originality that I make, is the giving them
proportion, place, design, and the shaping
them to a new utility. Thus I am rather
a compiler than a composer ; and I say
with Montaigne, ' I have gathered a nose
gay of flowers, in which there is nothing
of my own but the string which ties them.'
But in this case the string which ties
them is entirely my own.
' They (the Egyptian Priests) applied themselves
much to the study of Geometry and Arithmetic. The
Nile, which annually changed the aspect of the country,
gave rise to numerous lawsuits amongst neighbours,
with regard to the boundaries of their possessions.
These lawsuits would have been interminable without
the intervention of the Science of Geometry. Their
Arithmetic was useful in the administration of private
affairs and in geometrical speculations.' — DIODORUS.
THE USES AND TRIUMPHS
OF MATHEMATICS.
CHAPTER I.
INTRODUCTION.
' The oldest of the Sciences:
THERE is a certain Science which may
be compared to a series of mighty rivers,
several of whose sources — tiny springs
in some vast mountain — are inaccessibly
profound, — to a series of mighty rivers,
separate at first, and flowing in such diverse
directions that you would never guess of
any confluences ever occurring, but which,
suddenly bending round, join themselves
2 The Uses and Triiimphs of Mathematics.
to one another, forming thereby a stream
which, ever adding as it is new tributaries,
—a stream (to pass from Nature to the
Intellect) that under the name of MATHE
MATICS has given to the mind of man a
power, force, and rapidity in the investi
gation of Nature, increasing manifold its
capacities.
It is this that the history of Mathematics
teaches us. For from whence come our first
principles of this science — the oldest in the
world, and coeval, to some extent, with the
existence of man — we know not. And then
from what small beginnings has each de
partment of the science arisen : a Defini
tion, an Axiom, an Experiment. For, like
the other sciences, it existed, in observation
and practical applications, long before it
was established or reduced to the form of
a science by abstract reasoning. *
1 Pythagoras sacrificed a hecatomb when he discovered
the proof of Euc. I. 47.
Introduction. 3
Herodotus says : * I was informed by the
Priests of Thebes, that King Sesostris made
a distribution of the territory of Egypt
among all his subjects, assigning to each
an equal portion of land, in the form of a
quadrangle, and that from these allotments
lie used to derive his revenue by exacting
every year a certain tax. In cases, how
ever, where a part of the land had been
washed away by the annual inundations of
the Nile, the proprietor was permitted to
present himself before the king and signify
what had happened. The king used then
to send proper officers to examine and as
certain by exact admeasurement how much
of the land had been washed away, in order
that the amount of the tax to be paid for
the future might be proportional to the land
which remained. From this circumstance I
am of opinion that Geometry (the keystone
of Mathemetics) derived its origin, and from
thence it was transmitted into Greece/
4 The Uses and Triumphs of Mathematics.
And Plato also says, ' I have heard it
said that in the neighbourhood of Nau-
cratis, a town of Egypt, there had existed
one of the most ancient gods of this
country, who was named Theuth, and who
had invented the numbers, ciphering, geo
metry, astronomy, the games of chess and
of dice, and writing/
These were, of course, not all invented
by one man, nor yet at one period, but
this passage serves, however, to show their
vast antiquity. But it is not only from such
vague sources as the above that we derive
our knowledge of the Egyptian Mathema
tics, for if we see a people possessing
no mathematical knowledge before they
Note. — The British Museum preserves, under the name
of Papyrus de Rhind, the only treatise on Geometry that
Egypt has left us. This document dates from the XlXth
Dynasty, but it is, according to M. Birch, the copy of an
original which traces its origin back even to Cheops (B.C.
3091-67), i.e. about 5000 years ago. It is a very elementary
manual, containing a series of rules for the measurement of
surfaces and solids, presenting, at the same time, problems
for solution. — ' Les Premieres Civilisations Egyptiennes.'
Introduction. 5
have had relations with the Egyptians,
and possessing it very soon after these rela
tions have been established, it is assuredly
safe to infer that the first has borrowed
its knowledge from the second. Our first
o
principles of Geometry we can practically
trace back through the Greeks to the
Egyptians, our knowledge of Algebra (Arith
metic and Algebra were originally one, and,
in fact, until the end of the i6th cen
tury Algebra was little more than a con
venient shorthand for solving problems
in Arithmetic) we derive from the Arabs,
who transmitted it into Europe in the
loth or 1 3th century, and the Arabians
were pupils of the Hindu mathematicians,
Note A. — The Jesuit missionaries found very little know
ledge of Geometry amongst the Chinese. The Hindoos
possess a much larger amount of knowledge, but it is of very
uncertain date. No trace of any knowledge of Geometry is
found in the writings of the Jews.
NoteB. — Leonardo Bonacci (i3th century), a Pisan, whose
father was employed in the Custom House of Bugia, in
Barbary, acquired from the Arabs a knowledge of arithmetic
after the manner of the Indians.
6 The Uses arid Triumphs of Mathematics.
who were, in their turn, most probably
pupils of the Egyptians.
Thus we see that we owe the first
principles of this science (as indeed many
others) in all probability to the Egyptians.
Nevertheless, it is the Greeks whom we
have to credit with the real foundation of
the science of Geometry. Proclus says :
' That Pythagoras/ (about 600 B.C.) ' was the
first who gave Geometry the form of a
science/
The science was then greatly advanced
by the philosopher Plato, and the illustri
ous Euclid, whose ' Elements ' has been the
principal text book for beginners during a
period of more than 2000 years. Ptolemy
Lagus was one of his pupils, and it was
he to whom he made the celebrated reply,
when asked if there was no shorter way
to Geometry than by studying his ' Ele
ments,' — ' No, sire, there is no royal
road to Geometry.'
Introduction. 7
In the year 1619 Descartes, at the age
of twenty-three, by one of those extra
ordinary strokes of genius, occurring once
only in any age, fastened the irresistible
power of Algebra upon Geometry, thereby
giving to the mind a force and rapidity
in mathematical investigations quadrupling
its capacity. About this time also was
invented another analytical branch of Mathe
matics, known as the Calculus or the In
finitesimal Analysis.1 To explain the nature
of this analysis is not my object ; its power
and capacity are all I wish to mention.
Between the two methods — the geometrical
and the analytical — the following comparison
has been drawn : —
' Geometry had invigorated the reason,
as exercise toughens and strengthens the
Note. — Most of Sir Isaac Newton's great discoveries were
made before the age of twenty-seven.
1 Newton's ' Fluxionary Calculus,' 1666; Leibnitz's 'Dif
ferential Calculus,' 1667.
8 The Uses and Triumphs of Mathematics.
muscles of the human frame. But it had
given to the mind no mechanical power
wherewith to conquer the difficulties which
rose superior to its natural strength.
Archimedes wanted but a place whereon
to stand, and with his potent lever he
would lift the world. The student of
Physics demanded an analogous mental ma
chinery. What the human mind demands
and resolves to find out, it never fails
to discover. The Infinitesimal Analysis was
invented, its principles developed, and its
resistless power compelled into the ser
vice of human knowledge. So great is the
power of this analysis, that once having
seized on a wandering planet it never re
laxes its hold ; no matter how complicated
its movements, how various the influences to
which it may be subjected, how numerous
its revolutions, no escape is possible. This
subtle analysis clings to its object, tracing
its path and fixing its place with equal ease,
Introduction. 9
at the beginning, middle, or close of a
thousand revolutions, though each of them
should require a century for its accom
plishment.' 1
It must be added, however, that the
close and grasping character of the ancient
reasoning was lost ; but the time came
when Monge (the inventor of descriptive
Geometry) showed how to return to geo
metrical construction with means in many
cases superior to those of analysis in many
practical matters. The method of Monge
recalled the attention of geometricians to
the properties of Projection in general ; and
from the time of Monge to the present day
this subject (Geometry), has been cultivated
with a vigour which has produced re
markable results (notably in Electricity),
and promises still greater.
1 ' The Orbs of Heaven.'
Note. — The oldest work on Algebra now extant is that
of Diaphantos of Alexandria, in the 4th century after Christ.
io The Uses and Triumphs of Mathematics.
The history of the Applied Mathematics,
or Natural Philosophy, is equally interest
ing. It had also its foundation in experi
ment and practical application before it
existed as a science properly so-called.
The famous Archimedes was the great
founder of the sciences of MECHANICS and
HYDROSTATICS. Then Galileo, about the
latter end of the i6th century, greatly ad
vanced the science of MECHANICS, and his
pupil Toricelli that of HYDROSTATICS ; and
then the illustrious Newton, with his im
mortal discoveries in so many branches of
the science of Natural Philosophy.
ELECTRICITY was first observed by Thales
(600 B.C.), who noticed the property by
which yellow amber, on being subjected to
friction, attracted light bodies. But nothing
more was known of this science until the
close of the i6th century. The celebrated
mathematician Gauss (born 1777) invented
the magnetometer, and made MAGNETISM
Introduction. 1 1
an exact science ; and, in fact, may be re
garded as the founder of the truly scientific
study of magnetism.
Descartes (about 1620) brought the science
of OPTICS under command of Mathematics
by the discovery of the laws of refraction
through transparent bodies.
ASTRONOMY1 has always from the earliest
ages been intimately connected with the
science of Mathematics, and has always pre
sented problems to the mathematician not
only equal to all he could perform, but
passing beyond the limits of his greatest
intellectual power, and the solutions of many
of these problems are, indeed, wonderful illus
trations of the triumphs of mind over matter.
In the year 1788, Lagrange showed how
to apply mathematical analysis to mechani
cal problems, thus making the science purely
1 The first triumph of Mathematics would evidently be
the prediction of an eclipse ; the first prediction made by
whom, alas, unknown.
1 2 The Uses and Triumphs of Mathematics.
analytical. And since then the art of apply
ing Mathematics to the sciences of Electricity,
Heat, Magnetism, Hydromechanics, Optics,
etc., and, in fact, to every branch of Physical
Science, and the introduction of mechanical
principles into the theories of physical phe
nomena in general, has been most rapidly
and extensively cultivated, until at the pre
sent day Mathematics and Physical Science
have become in reality one. * For,' says
Professor Tyndall, ' no matter how subtle a
phenomena may be, whether we observe it in
the region of sense, or follow it into that of
the imagination, it is in the end reducible to
mathematical laws.'
Thus have I endeavoured, in a few words,
not to present you with a short history of
the science, but only to place before you a
few of its most striking and salient points ;
hoping by so doing to excite your interest to
pursue the subject further.1
1 See Montucla's ' Histoire des Mathdmatiques.'
Introduction. 1 3
Its first principles, under the head of
Arithmetic, are lost in the dim vista of the
past ; its origin rests on legendary ideas
alone. All we know is that numeration, or
the art of numbering, must have been to
some extent coeval with the existence of
man. Once transmitted into Europe, its
rapid progress and marvellous growth are
easily traceable. By the genius of different
mathematicians the various branches of the
science have been united, thereby giving to
the mind a power enabling it to pursue its
investigations with a force and rapidity in
creasing tenfold its capacity. And as the
various branches of the Pure Mathematics
have been invented and combined, so has it
been applied, both analytically and geometri
cally, more and more to the different branches
of science and art, and within the last half
century has attained such a state of perfec
tion as to enable a mathematician to deter
mine almost immediately whether a problem
14 The Uses and Triumphs of Mathematics.
can be solved by such means as he possesses
or not, — no small advantage, when it is con
sidered how much time was wasted in at
tempts to attain impossible solutions.
Applied to Engineering, it has enabled
man to bridge rivers and tunnel mountains ;
o
under the head of Electricity, it has enabled
him to ' Hash his words from the far land,
and girdle the earth with a spell ; ' and under
the head of that sublime subject, Astronomy,
its power is so great that should a star com
mence to revolve around some grand centre,
moving so slowly that millions of years must
roll away before it can complete one circuit,
not even a single year shall pass before its
motion be detected (by observation), in ten
years its velocity shall be calculated, and in
the lifetime of a single observer its period
shall become known. In a word, the astro
nomer, by observation and calculation, writes
out its history with perfect accuracy for a
million years.
Introduction. 1 5
These and other marvels not less wonder
ful perhaps justify the lines :—
' Some by it learnt the mysteries of the sphere,
The paths of comets, the movements of the stars,
The distajice of the sun : to some it gave
To prove th* existence of the self-same power
O'er time and space ; and law's unbroken reign,
And never varying energy : to others
But the arithmetician's lessened power,
Whilst others by its aid were led
' Neath mountains and o'er rivers?
The further contents of this little book is
an account of a few of its universal appli
cations and triumphs, and also a few words
about the inherent beauty and poetry of the
subject. We stand, so to speak, on the
verge of boundless possibilities ; what new
truths may be discovered, what new branches
of Mathematics may be invented, as Nature
becomes more and more disclosed, no man
can say. This further revelation of Nature
(God's work, and therefore His Word, if it
1 6 The Uses and Triumphs of Mathematics.
can be rightly interpreted) is man's highest
and noblest ambition.
The triumphs of mind over matter are
indeed most wonderful, and, at times, even
to the initiated, appear almost beyond man's
power. But man is born to aspiration as
the sparks fly upwards, and there is no
nobler or loftier ambition than
' To build in matter home for mind? 1
But let us always carefully remember the
words of two of the greatest mathematicians
that the world has hitherto seen — Newton
and Laplace ; the first of whom compared
himself to a child who had picked up here
a bright pebble and there a shining shell
on the shore, while the ocean of truth lay
all unexplored before him ; and the second
of whom said, — * What we know is a very
little {peu de chose], what we know not, is
immense.' And the illustrious mathema-
1 Emerson.
Introduction. 1 7
tician and philosopher, Pascal, wrote that
memorable sentence : — * The highest perfec
tion of human understanding- is to know
that there is an infinity of truth beyond its
reach.'
And the lesson of modern science has
been, in one sense, a negative one, for it
has revealed to man his utter insignificance
in the infinities by which he is surrounded,
and has taught us that first lesson we all
should learn — that of humble humility : —
. . . < Here
In this interminable wilderness
Of worlds, at whose immensity
Even soaring fancy staggers.' — SHELLEY.
' The Mathematics are either pure or mixed. To the
pure Mathematics belong those sciences which handle
quantity determinate^ l merely severed from any axioms of
natural philosophy. . . . The mixed Mathematics
hath tor subject some axioms or parts of natural philo
sophy, and considers quantity determined, as it is auxiliary
and incident unto them ; for many parts of Nature can
neither be discovered with sufficient subtility, nor ex
plained with sufficient perspicuity, nor accommodated
unto practice with sufficient dexterity, without the aid
and intervention of the Mathematics ; of which sort are
Perspective, Music, Astronomy, Cosmography, Archi
tecture, Engineering, and divers others.
' In the Mathematics I can discover no deficiency, ex
cept that men do not sufficiently understand the excellent
use ol the pure Mathematics, in that they do remedy
and cure many delects in wits and faculties intellectual.
For if the wit be dull, they sharpen it ; if too wandering,
they fix it ; if too inherent in the sense, they abstract it ;
so that as tennis is a game of no use in itself, but of great
use in respect that it maketh a quick eye, and a body
ready to put itself into all postures, so in the Mathematics
that use which is collateral and intervenient, is no less
worthy than that which is principle and intended. And
as for the mixed Mathematics, I may only make this pre
diction, that there cannot fail to be more kinds of them as
Nature grows further disclosed? — BACON.
i Viz., Arithmetic, Algebra, Geometry, Trigonometry, the Calculus,
Logarithms, Probabilities, etc., etc.
i\ott. — How wonderfully has his prediction been fulfilled, not
only with regard to the mixed Mathematics, but the pure also.
Under the head of the mixed Mathematics is included Mechanics,
Optics, Electricity, Heat, Astronomy, Pneumatics, Magnetism, and
also those continually calling in the aid of pure Mathematics, as
Geology, Geography, Geodosy, Land-Surveying, Navigation, Civil,
Practical, and Military Engineering, etc., etc.
CHAPTER II.
THE USES OF MATHEMATICS.
* Histories make men wise ; poets witty, the
Mathematics subtle, natural philosophy deep.'
BACON.
A SPECIAL interest that the history of the
science of Mathematics (hitherto unmen-
tioned) possesses, is the fact { that mathe
matical truths have always been referred to
by each successive generation of thoughtful
and cultivated men as examples of truth
and demonstration, and have thus become
standard points of reference among culti
vated men, whenever they speak of truth,
knowledge, or proof.' l
I now pass on to the consideration of the
1 DrWhewell.
The Uses of Mathematics. 2 1
subject-matter concerning which this Essay
was more especially written.
The utility of the science of Mathematics
in itself, and as a discipline of the mind,
lies in its strengthening the power of the
reasoning faculties by frequent examples,
which are the best lessons all can read,
and in its enabling anyone to distinguish be
tween reasonings founded on only probable
premises and on certain ones, and in forming
that habit known as concentration — of which
it has been said : — ' The one evil in life is
dissipation, the one prudence concentration ; '
and by means of which the greatest diffi
culties are overcome, and victory certain, if
only the right means be used ; and in cau
tioning anyone against receiving anything
which may appear at first probable enough
and based on sound reasoning, but which
when examined and analysed is seen to be
founded on false premises ; and in giving
to us a true and correct estimate of the
22 The Uses and Triumphs of Mathematics.
powers of the mind, by showing" the really
wonderful and varied consequences which
are able to be developed out of a few of
its most inherent notions ; and in giving
to us the pleasure of possessing a science
in which men of different nations, creeds,
and habits might a priori be expected to
agree. A knowledge of the pure Mathe
matics enables anyone to have all his know
ledge systematised and arranged. What
others have in confusion he will have in
order. The elements of knowledge are more
or less known to all, but in their most per
fect, communicable, and usable state they
are known only to a person possessing some
knowledge of Mathematics, if not of its
doctrines, at any rate of its methods. What
training is to the soldier, Mathematics is to
the thinker. Mathematics has conquered
Note. — The illustrious Newton when asked how he had
been able to achieve all his wonderful discoveries, replied, —
' By always intending my mind.'
The Uses of Mathematics. 2 3
contingency and verisimilitude, and shown
the fallacy of Chance.1
1 Chance/ says J. S. Mill, c is usually
spoken of in direct antithesis to Law ; what
ever (it is supposed) cannot be ascribed to
any law is attributed to chance. It is, how
ever, certain that whatever happens is the
result of some law ; it is an effect of causes,
and could have been predicted from a know
ledge of the existence of these causes, and
from their laws. If I turn up a particular
card, that is a consequence of its place in
the pack. Its place in the pack was a
consequence of the manner in which the
cards were shuffled, or of the order in which
they were played in the last game ; which,
again, were the effects of prior causes. At
every stage, if we possessed an accurate
knowledge of the causes in existence, it
would have been abstractedly possible to
foretell the event.'
1 Vide ' Chance and Luck,' by R. A. Proctor.
24 The Uses and Triumphs of Mathematics.
Further on, in his work on Logic, he also
says, ' Every event in itself is certain, not
probable, and if we knew all, we should
either know positively that it will happen
or positively that it will not. But the pro
bability to us means the degree of expectation
of its occurrence which we are warranted
in entertaining by our present evidence.'
It is this kind of chance or probability
with which Mathematics concerns itself.
Chance, then, as connected with Mathema
tics, has no connection with the ordinary
meaning of the word.
Mathematical Probability has shown the
fallacy of what is known as common consent,
and is a powerful auxiliary in the investi
gation or the discovery of some new law of
Note. — By means of the science of Mathematics is every
indirect measurement made. The utility of the science from
this point of view can hardly be over-estimated, because there
are many measurements which must be effected indirectly,
such as determining the distance and weight of the moon,
sun, planets, the velocity of light, electricity, the weight of
the earth, the distance of the stars, etc., etc.
The Uses of Mathematics. 25
Nature. Life in the aggregate is but a
Mathematical problem.
' Man,' said Jules Sandeau, ' has been
called the plaything of chance, but there is
no logic more close or inflexible than that
of human life ; all is entwined together, and
for him who is able to disentangle the
premises and patiently await the conclusion,
it is the most correct of syllogisms.'
But this is a digression. To attempt to
enumerate the different uses of each of
the various departments of the science of
Mathematics would be an impossible task
a task as wearisome to the general reader
as myself.
The uses of ARITHMETIC are known to
everyone : of them it would be absurd to
attempt an account. A man may be suc
cessful in business and be ignorant of Greek
or Latin, French or German, but unless
he have some knowledge, at any rate, of
Arithmetic, he certainly cannot be so. For
26 The Uses and Triumphs of Mithematics.
some years of boyhood there ought to be a
daily appropriation to the task of thoroughly
acquiring a perfect knowledge of the mani
pulation of this all - powerful instrument,
which by the new method (the unitary
method) is much less mechanical, and re
quires more exercise of the reasoning facul
ties than formerly.
1 The uses of GEOMETRY have always been
admitted, from the time of the Egyptians,
who settled their lawsuits by means of it,
and Plato, who placed the following inscrip
tion over the door of his house : ' Whoso
knows not Geometry, let him not enter
here,' down to the present day, when the
Parisian dressmakers are taught in the pro
fessional schools of the city of Paris not
only sewing but Euclid or Geometry and
Drawing. Geometry strengthens and invi-
1 Vide Chalmers' ' Graphical Determination of Forces in
Engineering Structures,' Sir W. Thompson's Papers on
* Electrostatics and Magnetism,' and Clerk Maxwell's ' Elec
tricity.'
The Uses of Mathematics. 2 7
gorates the reason, as exercise toughens and
strengthens the muscles of the body ; and
Lord Bacon's statement, ' that if a man's
mind be wandering, let him study the
Mathematics/ applies to no branch of that
science more than to Geometry (notably
1 The Elements of Euclid '). Once lose the
chain of reasoning, it is no use ; you must
go back to the beginning of the proposi
tion, and begin again, if you wish to under
stand what you are reading.'
And pure Geometry must ever remain
the most perfect type of the deductive
method in general. ' And the recollections
of the truths of pure Geometry has, in all
ages, given a meaning and a reality to the
best attempts to explain man's power of
arriving at truth.'
The practical uses of Trigonometry (a
combination of Algebra and Geometry),
under the head of land surveying, geo-
dosy, etc., are well known. And what is
28 The Uses and Triumphs of Mathematics.
known as Spherical Trigonometry is of
great use in Nautical Astronomy, to which
navigation, and therefore commerce, owes
so much.
I now pass on to the uses of ALGEBRA,
LOGARITHMS, the CALCULUS, PROBABILITIES,
and the higher branches of the pure Mathe
matics. The uses of these are not so well
known, because, with the exception of
a slight knowledge of Algebra, by far
the greater portion of mankind have no
knowledge of these subjects, and their
uses, vast though they be, are known only
to a few.
One of their many uses is an immense
saving of time and labour ; problems which
by more elementary means would require
sheets of paper and days of labour, are
solvable in a few lines and in a few
minutes. And as this holds for even the
elementary portions of Algebra over or
dinary arithmetical calculations, so does it
The Uses of Mathematics. 29
hold proportionately with regard to the
Calculus over Algebra and Geometry. Pro
blems which, without the Calculus, require
much thought and labour, and often a great
deal of ingenuity for their solution, can be
solved systematically by the Calculus, with
out any need of ingenuity, so long as the
proper rules are followed. By the higher
Mathematics are solved of course problems
not solvable by elementary means, however
much ingenuity be used ; and many of the
results obtained by the Calculus have their
practical applications in the rules used in
Mensuration, for instance. But it is not
these uses to which I wish particularly to
call your attention. Mathematics has other
uses besides the determining of heights and
distances, the finding the volumes of solids,
and the like ; or
' By geometric scale
To take the size of pots of ale.' l
1 Thomas Carlyle.
30 The Uses and Triumphs of Mathematics.
THE REAL PRACTICAL USE of the science
of Mathematics lies in its application to
the sciences of Mechanics, of Optics, of
Acoustics, of Hydro-Mechanics, of As
tronomy, of Electricity, of Magnetism, of
Heat, of Chemistry, of Geology, of Biology,
of Engineering, of Music, of Architecture,
of Painting, of Pneumatics, of Navigation,
and, in fact, directly or indirectly, to the
whole domain of Science and Art ; ' from
investigations relating to the infinitely great
and the infinitely little to the study of the
most familiar objects of every-day life.'1
And what are the uses of Science ? Fara
day answered this question by demanding :
' What was the use of a baby ? ' But, apart
from this, the use of Science, is this — man
kind has at last realised the fact that
nothing happens by accident, and that there
is no such thing as chance. Thus, nothing
happening by accident but by Law, it be-
1 R. A. Proctor.
The Uses of Mathematics. 31
hoves us to become acquainted with these
laws, in order that we may guide our
practical conduct by them. This it is the
aim of Science to achieve. We owe all
our knowledge of Nature to Science.
Hence its use. It was not for this — as an
instrument in the discovery of the laws of
Nature — that the science of Mathematics
was valued by many of the ancients. They
valued it only ' as leading men to the
knowledge of abstract, essential truth.' l
Archytas framed machines of extraordinary
power by mathematical principles. Plato
remonstrated with him, and declared that
this was to degrade a noble intellectual
exercise into a low craft, fit only for car
penters and wheelwrights. The office of
Geometry was to discipline the mind, not
to minister to the base wants of the body.
This interference was successful, and from
that time, the science of Mechanics was
1 Plato's ' Republic,' Bk. 7.
32 The Uses and Triumphs of Mathematics.
considered as unworthy of the attention of
a philosopher. And even Archimedes was
not free from the prevailing notion that
Geometry was degraded by being employed
to produce anything useful. It was with
difficulty that he was induced to stoop
from speculation to practice. He was half
ashamed of those inventions which were the
wonder of hostile nations, and always spoke
of them slightingly as mere amusements, as
trifles in which a mathematician might be
supposed to relax his mind after intense ap
plication to the higher parts of the science.1
With increased knowledge has come in
creased wisdom, and we now value Mathe
matics as the handmaid of Science. There
is no doubt beauty in the idea that ' The
soul, considered in relation to its Creator,
is like one of those mathematical lines that
may draw nearer to another for all eternity,
without a possibility of touching it ; ' or in
1 See Lord Macaulay's ' Essays,' Lord Bacon.
The Uses of Mathematics. 33
comparing ' the directrix or axis of a curve
which extends both ways to infinity, with
out ever deviating to the one side or the
other, to the infinite and unbending recti
tude, truth, and justice of the great Creator.'
But it is not for illustrations such as these
that we desire to become acquainted with
the conic-sections.1
The real use of the Science of Mathe
matics lies in its applications, and they
may be said to be universal. From deter
mining the stability or non-stability of a
ship, to determining the stability or non-
stability of the planetary system ; from
calculating the path of a projectile, to
calculating the path of a comet ; from
finding the cubical contents of an ordinary
wall, to finding the cubical contents of the
sun ; from computing the distance of an
object a mile off, to computing the distance
}f a star at a distance of billions of miles ;
1 Vide Chap. V.
C
34 The Uses and Triumphs of Mathematics.
from estimating the weight of a few tonsr
to estimating the weight of the whole solar
system ; from calculating the velocity of a
railway train, to calculating the velocity
of light, with a velocity of over 180,000
miles per second ; in a word, from the
ordinary money or business transactions
of everyday life, up to the most elaborate
calculations of the student of Physics, the
Science of Mathematics is of the greatest
possible kind of use.
But, as Professor Tyndall so aptly puts
it : ' The circle of human nature is not
complete without the arc of feeling and
emotion. And here the dead languages,
which are sure to be beaten by science in
a purely intellectual fight, have an irresist
ible charm. They supplement the work
of Mathematics, by exalting and refining
the aesthetic faculty, and must be cherished
by all who desire to see human culture
complete.'
The Uses of Mathematics. 35
To omit one is to leave a man half-
educated. To crarn into a man a certain
amount of knowledge concerning the mani
pulation of certain symbols is not to edu
cate him at all ; in order that we may
share in what men are doing in the world,
we must share in what they have done.
Thus arises the importance of history. As
refining the aesthetic faculty, the Classics
and Fine Arts are invaluable, and their
claims must not be set aside.
And manifold and varied though the
uses of Mathematics be, we must not
let those uses become abuses ; for, in
our hours of pleasure and enjoyment, we
are quite right to say with the poet
Shelley, —
' As to nerves,
With cones, and parallelograms, and curves,
Pve sworn to strangle them if once they dare
To bother me — when you are with me there,'
and, above all, let us take care of our
36 The Uses and Triumphs of Mathematics.
health ; it is better to cultivate the body
at the expense of the mind than the mind
at the expense of the body, for
1 Health is the first wealth?
'Those long chains of reasoning, all simple and
easy, by which Geometers used to arrive at their most
difficult demonstrations, suggested to me that all things
which come within human knowledge must follow
each other in a similar chain, and that, provided we
abstain from admitting anything as true which is not so,
and that we always preserve in them the order necessary
to deduce the one from the other, there can be none
so remote to which we may not finally attain, nor so
obscure but that we may discover them.' — DESCARTES.
CHAPTER III.
THE TRIUMPHS OF MATHEMATICS.1
1 Mathematicians' art is ever able
To endow with truth mere fable?
I.
UNIVERSAL GRAVITATION.
' Nature and Nature's law lay hid in night,
God said, "Let Newton be" and all was light?
IN giving a short account of a few — I may
say a very few — of the Triumphs of Mathe
matics, I will first mention the discovery,
or rather the demonstration, of the law of
Universal Gravitation, as given by the il
lustrious Newton in that wonderful book
' The Principia/ — of that grand law, that
mighty power, that mysterious hand, so to
1 Vide ' Mecanique Celeste' (Laplace), ' Mecanique Analy-
tique' (Lagrange), and Euler's ' Scientific Papers.'
4O The Uses and Triumphs of Mathematics.
speak, which causes bodies when unsup
ported to fall to the ground, the moon to
describe its orbit about the earth, the tides
to perform their daily ebb and flow, the
planets to revolve around the sun, floating
isolated in space, and the whole solar system
to revolve around some grand centre, de
scribing an immense but purely ideal curve,
existing on in theory and in the decree of
eternal laws, — of that law which, as far
as we know, exists everywhere, which has
created order out of chaos, and which holds
the universe together.
That a law existed had been suspected
long before the time of Newton ; and it
had even been conjectured ' that it varied
inversely as the square of the distance/
Note. — The distinct part of Newton's great discovery was
not the motion of attraction, which had occurred to many
of the ancients, — not the law which had been suggested by
Kepler and Bouillard, but the proof that the mechanical
deductions from this law of attraction did really represent
observed phenomena.
The Triumphs of Mathematics. 4 1
The manner in which Newton established
his theory of Universal Gravitation was as
follows. He first considered the moon as
a body falling,1 in one sense, towards the
earth, — that is, seeing if the rate at which it
fell towards the earth agreed with the laws
of falling bodies as propounded by Galileo.
Newton first computed, from the known
velocity of the moon in its orbit, and from
the radius of that orbit, the distance
1 What Is meant by the moon A
falling towards the earth, is sup
posing the curved line A D C to
represent the path of the moon X |c
from A to C. The point E being
the situation of the earth, A that
of the moon, then were it not for
the attraction of the earth the
moon would proceed along the
straight line A B, and traverse
A B in the same time that it would 0
have taken to go from A to C K
along the curved line A U C under the earth's attraction ;
thus supposing AD C to be the distance the moon goes
in one second of time, B C is the distance, so to speak,
through which it has fallen in that period, due to the earth's
attraction.
42 The Uses and Triumphs of Mathematics.
through which the moon actually fell towards
the earth in one second of time. He next
computed the distance through which a
heavy body would fall towards the earth's
surface, if removed to the distance of the
moon from the earth's surface. Now if
these two quantities were equal, then the
truth of his demonstration (as far as the
moon and earth are concerned) was com
plete, because, if so, the moon did fall
through that distance required by the
assumed law, and therefore this law (for
earth and moon) was a law of Nature. This,
after the greatest labour and calculation, he
found to be the case. Having thus estab
lished his great theory in the case of the
moon, he next proceeded to establish it for
the planets and solar system generally. This
he accomplished by means of certain pro
positions, demonstrated in his ' Principia,7
and three famous laws with regard to the
sun and planets, known as Kepler's Laws,
The Triumphs of Mathematics. 43
from the name of the discoverer, who dis
covered them after years of continuous
observations of the sun and planets, and
laborious mathematical calculations.
Having thus demonstrated his law for the
Solar System generally, Newton was led on
to infer his grand theory of Universal Gravi
tation, which is as follows : — Every particle
in the universe attracts every other particle
with a force of attraction in the line joining
them, proportional directly to their mass ;
and proportional inversely to the square of
the distance between them.
Newton was only able to demonstrate his
law with regard to the Solar System, but
his inference modern science has fully con
firmed, and has but infinitely increased our
ideas of the marvellous wonders and powers
of this mysterious law. The illustrious
French mathematician Laplace has shown
that the velocity of the action of this law,
or of gravity, must be several, if not many,
44 The Uses and Triumphs of Mathematics.
millions of times greater than the velocity
of light, and the velocity of light is over
180,000 miles per second. And as its rate
of propagation is infinitely great, so is the
distance through which it acts. It annihil
ates both space and time. This mystery of
the universal action of Gravity is the greatest
of all modern or ancient scientific marvels ;
and the deeper we go into it the deeper
grows the mystery. If Light, Electricity,
etc., be but modifications of the action of
Gravity, that renders nothing more simple,
but only infinitely more wonderful. It is
the mystery of mysteries, and seems to be
almost in some sense associated with the
great First Cause.
Attempting its unravelment : —
* Charmed and compelled thou climVst from height to height,
And round thy path the world shines ivondrous bright,
Time, Space, and Size, and Distance cease to be,
And every step is fresh Infinity? x
1 Goethe.
The Triumphs of Mathematics. 45
n.
RE-DISCO VER Y OF THE ASTEROID CERES.
1 Oh thou small fragment of a world once
Beautiful and bright and fair as this is, till
WrecKt in some, convulsion. Oh float
Into our azure sky once more?
THE next triumph I will relate is the
re-discovery of the Asteroid Ceres after it
had become lost in the rays of the sun,
owing to its having been discovered by
Piazzi in such a position that he was able,
on account of illness, only to make a few
observations of it prior to its being lost in
the rays of the sun. What a hopeless task
it seemed, its re-discovery, as the telescope
would have to grope its way around the
heavens, slowly and carefully, in that region
known as the Ecliptic, comparing every
star with its place in the chart or catalogue
of stars. What was to be done ? In this
dilemma mathematical analysis attempted
to create an orbit for this lost planet by
46 The Uses and Triumphs of Mathematics.
means of the data afforded by the few
observations which Piazzi had been able
to make. What were these data ? It had
been observed during its passage over an
arc of 4° out of 360°, approximately taken
the orbit as circular. What an absurd
attempt, do you say ? You know not the
powers of this wonderful analysis. The
genius of the great mathematician Gauss,
then quite young, succeeded in this her
culean task, and when the telescope was
pointed to the heavens in the exact spot
indicated by this daring computer, there,
in the field of view of the telescope, shone
the delicate and beautiful light of the long-
lost planet. This was indeed a wonderful
triumph of analytical skill and reasoning,
and another verification of the saying, ' That
fiction can never be more wonderful or
superior to truth ; ' the latter is indeed a
source of inspiration to us, richer and more
enduring than the former.
The Triumphs of Mathematics. 47
in.
THE DISCO VER Y OF NEPTUNE.
''Hence the view is profound,
It floats between the world
And the depths of the sky' — GOETHE.
THE astronomer M. Bouvard of Paris, in
the year 1820, prepared tables by means of
which the motions of three great planets,
Jupiter, Saturn, and Uranus, might be pre
dicted. The positions of the two planets,
Jupiter and Saturn, were found to agree with
those predicted, and their motions with the
theory of Gravitation. But not so in the case
of the planet Uranus. In a few years Uranus
began to deviate from the places indicated,
and in the year 1844 the error amounted to
four minutes, or about one-seventh the dia
meter of the moon, — a very small quantity,
from a non-astronomical point of view, but un
able to be overlooked by this the exactest of
the sciences. Nor could it have been over
looked if it had been an eighth part of that
48 The Uses and Triumphs of Mathematics.
amount. Analogy suggested that these dis
crepancies must be due to the attraction of
some unknown planet. This planet evi
dently could not be between Saturn and
Uranus, for then Saturn would have been
affected as well as Uranus. Thus the orbit
of this unknown planet must be outside that
of Uranus. Two young mathematicians, the
one English, and the other French, whose
names were respectively Adams and Le
Verrier, independently and qinte unknown to
each other, undertook this apparently super
human task of discovering the new planet, be
ing given the perturbations of Uranus. And
there is this difference between this task and
the discovery of the Asteroid Ceres, in the
case of Ceres the planet was known to exist,
and had even been observed, although only
for a very short time, but in the case of
the discovery of the planet Neptune, no
eye had ever beheld it, i.e., as a planet, and
about any of its elements or data nothing
The Triumphs of Mathematics. 49
was of course known ; these it was the busi
ness of the two mathematicians to discover,
by means of Newton's theory of Gravitation,
and Mathematical Analysis.
Mr Adams began his calculations and in-
o
vestigations in 1843, a°d in October 1845
he communicated the results of his calcula
tions to the Astronomer-Royal, and to Pro
fessor Challis of Cambridge University in
August 1846. Professor Challis found the
planet, but, under pressure of other business,
did not recognise it. In the meantime, M.
Le Verrier, at the instigation of M. Arago,
O O '
had investigated the problem, and commu
nicated his results to the French Institute
in November 1845, June 1846, and August
1846; and on 25th of September 1846,
Dr Galle, assistant to Professor Encke, of
Berlin, discovered the new planet, from a
communication which he had received from
Le Verrier. What is known as the Helio
centric position of the planet as found by—
D
50 The Uses and Triumphs of Mathematics.
Dr Galle was 326° 12'.
As compared by Mr Adams, 329° 19'.
As computed by M. Le Verrier, 326° o'.
Thus the real discovery of the planet was
due to M. Le Verrier and Dr Galle, though
this does not, of course, in any way detract
from the fame and merit due to Mr Adams
in the undertaking and so successfully solv
ing so grand a problem. From the above
we see that the computation of M. Le Verrier
was rather more accurate than that of Mr
Adams, but Mr Adams was of course quite
correct enough for all practical purposes.
And this double calculation must demonstrate
that the position of the planet as assigned
by the two computers was not one of mere
chance, but that it was one determined
by means of pre-eminent ability and skill,
based on sound principles, and approached
by an accurate and logical process. And in
this respect these two great mathematicians
are not rivals, but vindicators of each other,
The Triiimphs of Mathematics. 5 1
IV.
PREDICTIONS OF THE RETURN OF
COMETS.
' That mysterious visitant whose beauteous light
Among the wandering stars so strangely gleams !
Like a proud banner in the train of night,
Th emblazon! d flag of Deity it streams —
Infinity is written on thy beams :
And thought in vain would through the pathless sky
Explore thy secret course. Thy orbit seems
Too vast for Time to grasp. Oh, can that eye
Which numbers hosts like thee, this atom Earth descry ? '
' Les Merveilles Celestes.'
Two thousand years ago Seneca wrote :—
* A day will come when the course of
these bodies (comets) will be known, and
submitted to rules like those of the planets.'
The prophecy of the philosopher has been
fulfilled. Thought ' has explored their
secret courses ; their orbits are not too
vast for man to grasp.' The comets, like
the planets, obey Newton's law of Universal
Gravitation, and are subject to all its varied
52 The Uses and Triumphs of Mathematics.
influences. The first prediction of the re
turn of a comet was made by an English
man, viz., the illustrious Halley, which
return he knew he himself would never
be able to behold. This comet (Halley's)
appeared in 1682, and Halley studied it
with great care and attention ; and after
great labour he computed the elements of
its orbit, and found it to be moving in an
ellipse of great elongation, i.e., greatly ex
tended or flattened out, so to speak, and
that it receded from the sun to a distance
of 3,400,000,000 miles. And he predicted
its return about the close of 1758 or the
beginning of 1759. The first glimpse
caught of it was by G. Pabtch, an ama
teur peasant astronomer, on December 25th,
1758, returned to crown with glory the
English mathematician and astronomer who
had predicted its return after an absence
of seventy-six years.1
1 The return of this comet was computed to within a period
The Triumphs of Mathematics. 53
The next return of this comet was com
puted to within nine days of its actual
occurrence — a most remarkable calculation,
since it never escapes from the attractive
influence of the planet Neptune, even
when at its furthest distance from the
sun.
You may say, but all the ' Triumphs '
which you have related, so far, are taken
from the subject of Astronomy. Has
Mathematics achieved no triumphs in any
other department of science ? It has, in
every department of Physical Science. It
has triumphs, no less nobly achieved, in
the sciences of Electricity, Mechanics,
Optics, and, in fact, in the whole range of
Physical Science.2 I will take one from
of nineteen days of its actual occurrence by two French
mathematicians, Lalande and Clairvaut, assisted by Madame
Lalande, they allowing themselves thirty days either way,
on account of their neglecting small irregularities. The
disturbing influence of Neptune was of course then unknown.
2 Vide ' The Cambridge and Dublin Mathematical Jour
nal,' ' The Philosophical Magazine,' etc.
54 The Uses and Triumphs of Mathematics.
the science of Optics, both recent and im
portant, viz. : —
v.
THE DISCO VER Y OF CONICAL
REFRACTION.
' First, mathematicians skill,
And after, keen opticians gaze .
Explored the doctrine of those rays?
THE mathematician Fresnel had calcu
lated the mathematical expression for the
wave surface in crystals possessing two optic
axes, but he did not seem to have any
idea of refraction in such, except a refrac
tion known as double refraction. Sir Wil
liam Hamilton, of Dublin, the inventor of
a mathematical method known as Quater
nions, and a most profound mathematician,
took the subject up at this point, and
proved that the theory known as the Undu-
latory Theory of Light pointed to the con-
The Triumphs of Mathematics. 55
elusion that at four special points of the
wave surface the ray of light was divided
not into two but into an infinite number
of parts, forming, therefore, at those four
points, a continuous conical envelope or hollow
cone, instead of two images, as had been
hitherto supposed. No human eye had
ever seen this conical envelope when Sir
William Hamilton said it existed, any more
than any eye had ever beheld the planet
Neptune until Mr Adams and M. Le
Verrier demonstrated its existence ; both
were previously ideas or theories in the
minds of mathematicians. Dr Lloyd took
a crystal of a mineral known as Arragon-
ite, and following with scrupulous exact
ness the indications of Sir William Hamil
ton's theory, he discovered this wonderful
envelope.1
You may say these triumphs are indeed
most wonderful, almost incredible, and truly
i See ' Notes on Light,' by J. Tyndall, F.R.S.
56 Tke Uses and Triumphs of Mathematics.
prove ' the potent power of mind o'er
matter ; ' but they, or, at any rate, some
of them, scarcely appear to be of much
practical use.
A few words, then, as to their uses. For
this is the chief reason why I chose the
above.
Newton's discovery of the law of Uni
versal Gravitation remodelled and vastly im
proved the whole science of Mechanics ; it
gave us the true theory of the movements of
the heavenly bodies, and became the parent
of innumerable other discoveries.
Navigation, and, therefore, Commerce and
Industry, immediately felt its influence, and
every individual of our species has derived,
and will continue to derive, as long as man
kind exists, incalculable benefits therefrom,
both intellectual and material.
The discovery of the planets Neptune and
Ceres was a consummate verification of the
law of Universal Gravitation, just as the
The Triumphs of Mathematics. 57
discovery of Conical Refraction was a con
summate verification of what is known as
the Undulatory Theory of Light ; these
discoveries amounting to almost absolute
proofs of two of the grandest and most
useful theories ever propounded.
Comets were considered by the ancients,
and in the Middle Ages, as objects of terror,
—miraculous apparitions, forerunners of aw
ful calamities, burning symbols of Divine
wrath. But now, thanks to the labours of
mathematicians and observers, these bodies
(as we have seen) are regulated by and sub
ject to the same laws as the planets. They
have been robbed of their terrors, and are
regarded by Schiaperelli and others as
1 In Roman history there is a remarkable story of a
Roman nobleman, an astronomer and mathematician, who,
when he was serving against the Macedonians, under Julius
/Emilius,_/tfn?/<?/^ to the Roman soldiers an eclipse, and ex
plained its causes, and thereby preventing the consternation
they otherwise would have fallen into, and which, seizing
their enemies, they were easily routed by the Romans. —
Guithric.
58 The Uses and Triumphs of Mathematics.
analogous to meteors, — bodies fleeing from
world to world, scattering in their course in
the neighbourhood of the stellar systems the
dust of the elements of which they are com
posed — carbon, and perhaps hydrogen ; car
bon, which is such an important factor in life,
thus preserving perhaps life on the surface
of those planets on which it falls.
And with regard to the practical use of
certain other triumphs, it must be remem
bered that there are many great discoveries
which, though they may appear at first sight
of little use in themselves, have given birth
to others of the greatest utility. When these
results can be practically used in the increase
of the mass of general knowledge and wealth,
then is their use at once perceived. But of
greater use (though perhaps unperceived) are
those prior discoveries which led up to them ;
for without the first the second could have
had no existence. By the existence and
assistance of the higher branches of pure
The Triumphs of Mathematics. 59
Mathematics are those sciences to which they
are applied rendered more powerful, perfect,
and of greater service to man ; and by the
aid of Electricity, Navigation, Engineering,
Geodesy, etc., is commerce and industry
vastly improved ; and thus is every indivi
dual incalculably benefited — indirectly, it may
be, but none the less so on that account.
And, moreover, these and other triumphs in
every department of science ' have carried
us to sublime generalisations — have affected
an imaginative race like poetic inspirations.
They have taught us to tread familiarly on
giddy heights of thought, and to wont our
selves to daring conjectures.' And they have
also revealed to us the infinities by which we
are surrounded. They have taught us to
see a system in every star, but they have
Note. — By the recent elaborate mathematical investiga
tions of Sir William Thompson, Clerk Maxwell, etc., has the
science of Electricity been entirely changed, and calcula
tions can now be made with regard to electrical phenomena
with as much certainty as calculations in dynamics. — See
' Electricity in the Service of Man.'
60 The Uses and Triumphs of Mathematics.
also taught us to behold a world in every
atom, the one teaching us the insignifi
cance of the world we tread on, the other
redeeming it from every insignificance.
Eager and ever curious, man presses onward
for the accomplishment of further triumphs,
the solution of grander problems. What we
know is as nothing to what we know not.
The solution of these problems is man's
highest and noblest ambition ; and there is
no truer truth than that—
' Nature when she adds difficulty adds brain.''
EMERSON.
' It is an error to ascribe discoveries to Mathematics.
It happens with this, as with a thousand other things,
that the effect is confounded with the cause. Thus
effects which have been ascribed to the steam-engine,
belong properly to fire, to coals, or to the human mind.
The true discoveries in Mathematics are successive steps
towards the perfection of the instrument, by which it is
rendered capable of innumerable useful applications,
but Mathematics alone makes no discoveries in Nature.'
—JUSTUS LIEBIG, f Letters on Chemistry.'
CHAPTER IV.
THE LIMITS OF MATHEMATICS.
' Mathematical Science is the handmaid of
Natural Philosophy.' — BACON.
IT is perfectly true that Mathematics of it
self makes no discoveries in Nature, and
that, besides Mathematics, a high degree of
imagination, acuteness, and talent for obser
vation are required to make discoveries in
Physical Science. The imagination has al
ways been a powerful factor in the discovery
of Nature, and without observation we can
know nothing. ' Observation of Nature is
the only source of truth.' ' Experiment is
invented observation.' It is the duty of the
philosopher to explain and illustrate the facts
64 The Uses and Triumphs of Mathematics.
of Nature by experiments. * No single iso
lated phenomena, taken by itself, can furnish
us with its own explanation ; it is by tracing
its consequences, by studying and arranging
its antecedents and consequents, and well
observing their several links, that we attain
to a comprehension of it, and an understand
ing of its true cause. For we must never
o
forget that every phenomenon has its reason,
every effect its cause'
And at this point Logic and Mathematics
take up the subject, the one (Logic) to verify
that which the imagination, with its far dart
ing glance, has seen, for from the moment
the imagination is allowed to solve questions
left undecided by researches, investigation
ceases, truth is unascertainable, and in error
is created a MONSTER — envious, malignant,
and obstinate — which, when at length truth
endeavours to make its way, crosses its path,
combats, and strives to annihilate it ; and
the other (Mathematics) to reduce the phe-
The Limits of Mathematics. 65
nomena to mathematical laws for future use.
and as a verification of his experiments ; for
if his calculations agree not with his experi
ments, then the conditions of an accurate
and logical process are not satisfied, and no
discovery has been made.
Thus Mathematics, though the last, is by
no means the least y factor in the discovery.
And it must be remembered that all the
sagacity, acuteness, and talent in the world
would be useless without the instrument.
The instrument might be created, you say.
Exactly what has been done, but this de
tracts not either from its power or use. A
steam-engine is none the less important be
cause it is only a steam-engine.1
An erroneous and rather curious idea
often entertained with regard to the higher
1 The order indicated above, of course, is not always
followed. In the discovery of Conical Refraction, for in
stance, Mathematics came first, Observation afterwards.
The above is the most unfavourable case, so to speak,
Mathematics being not so much the discoverer as verifier,
though one is useless without the other.
66 The Uses and Triumphs of Mathematics.
branches of Elementary Mathematics is that
it is possible to prove all manner of incon
gruities by means of them, — that two is equal
to four, and many such like absurdities.
This, of course, is not so. This and other
illustrations are simply examples of what
absurd results may be arrived at if our
data be incorrect, our reasoning false, or
some element or factor left out in the cal
culations. It was proved by the Fluxionary
Calculus that steamships could never get
across the Atlantic. But in spite of the
Fluxionary, or any other Calculus, this has
been done. But this does not of necessity
denote, as some would suppose, the falsity
or weakness of the Fluxionary Calculus, but
points to false data, incorrect reasoning, or
unknown elements left therefore out of con
sideration. Another erroneous idea is that
the results of the calculations of the higher
Mathematics are only approximations, and
not exact. With regard to this, the illustri-
The Limits of Mathematics. 67
ous Carnot has said, — ( The important-
one may say the sublime — value of the In
finitesimal Analysis is in joining with the
facility of the process of a simple approxi
mate calculation, the exactitude of the re
sults of ordinary analysis. . . . The
objections made against it (the Infinitesimal
Analysis) all rest on the false supposition
that the errors committed in the course of
the calculation, in neglecting the infinitely
small quantities, remain in the results of
that calculation, however small one may
suppose them to be. But this is not so :
the elimination takes them all away.' Thus
we see, as every student knows, that the
results of the higher Mathematics are not
approximations but exact.
But the science of Mathematics deals with
the physical and not with the ideal. It is
an instrument constructed by man for the
use of man, and of necessity, therefore, like
all of man's creations, is far from perfect,
68 The Uses and Triumphs of Mathematics.
and has its narrow limits. I believe I am
right in saying that no mathematician is able
to calculate the exact curve which a tossed
penny describes, the influences to which it
is subjected being so numerous and inter
mixed in such a complex manner with one
another. And there are problems whose
solution, even supposing the laws of the
influences to which they are subjected being
known, the computation of their aggregate
effect appears to be beyond the powers of
the Mathematical Analysis as it is or is
ever likely to be. Thus this instrument,
powerful as it is, has its limits.1
And it has also its limits in another way,
namely, the limits of its applicability to the
improvement of the other sciences. It is
not difficult to conceive how chimerical would
be the hope of applying mathematical prin
ciples to some of the complex inquiries of
such subjects as physiology, society, govern-
1 See Comte's * Positive Philosophy,' vol. iii.
The Limits of Mathematics. 69
ment, etc. But the failure of the science
even here is only partial. Its principles
may fail but its methods be still applicable.
' The value of mathematical instruction as
a preparation for those more difficult in
vestigations (physiology, society, govern
ment, etc.) consists in the applicability, not
of its doctrines, but of its methods.
( Mathematics will ever remain the most
perfect type of the Deductive Method in
general ; and the applications of Mathe
matics to the simpler branches of physics
furnish the only school in which philosophers
can effectually learn the most difficult and
important portion of their art, the employ
ment of the laws of simpler phenomena for
explaining and predicting those of the more
complex.
' These grounds are quite sufficient for
deeming mathematical training an indispens
able basis of real scientific education, and
regarding, with Plato, one who is ageometre-
7O The Uses and Triumphs of Mathematics.
tos as wanting in one of the most essential
qualifications for the successful cultivation
of the higher branches of philosophy.' 1
1 J. S. Mill, 'Logic,' vol. ii. p. 180.
' Beauty chased he everywhere
In flame, in storm, in clouds of air.
He smote the lake to feed Jiis eye
With the beryl beam of the broken wave ;
He flung in pebbles well to hear
The moments music which they gave.
Oft pealed for him a lofty tone
From nodding pole and belting zone.
He heard a voice none else could hear
From centred and jrom errant sphere.
The quaking earth did quake in rhyme,
Seas ebbed and flowed in epic chime? — EMERSON.
CHAPTER V.
THE BEAUTY OF MATHEMATICS.
'There can be no Beauty where Chaos reigns.'
' BEAUTY,' said a philosopher, ' possesses that
which is simple ; which has no superfluous
parts ; which exactly answers its end ; which
stands related to all things ; which is the
mean of many extremes.'
This definition applies equally well to the
science of Mathematics. In Mathematics all
our knowledge is in the number of primi
tive data or conclusions which can be drawn
therefrom. Around these first principles as
around a standard everything associates ; no
matter how remote it may be, it unites itself
to that to which it has been well attached.
Every theorem in each department of the
74 The Uses and Triumphs of Mathematics.
science has a kindred connection with every
other theorem. What has been so exqui
sitely sung of the associations of childhood,
is true (altering the connection only) of the
associations of the different departments, etc.,
of Mathematics. For in Mathematics —
' Up springs at every step, to claim a place,
Some little AXIOM making sure the '•pace ' ;
And not a PROBLEM — but what truly teems
With golden visions and romantic dreams.'1
The language of Mathematics has its own
peculiar beauty. Symbolical as it be, it is
also symbolical in another sense.
As the artist has to employ symbols to
give us either the spirit or the splendour of
Nature, and to convey his enlarged sense to
his fellow-men, and to open men's eyes to
the mysteries of eternal art, so the mathe
matician is compelled to use symbols to
clothe his art in a language enabling him to
best convey his enlarged sense with brevity
and clearness to others, and to enable him
The Beauty of Mathematics. 75
also to open men's eyes to the powers of the
human mind, and to grapple so successfully
with those problems pressing for solution on
every hand. It is the wonderful simplicity,
power, and utility of these symbols which
constitute their beauty.
The beauty of Mathematics is not that
of a pageant or ballet. There are many
beauties, — moral beauty, beauty of manners,
of the human face and form, of the intellect,
of Nature, and of that which enables us to
understand Nature, and discover her laws.
It is this latter beauty that the science of
Mathematics possesses.
There is an ascending scale of the per
ception of beauty, from the joy which some
grand spectacle affords the eye, to the per
ception that symmetry of any form is beauty
up to perception of the Goethe, that the
beautiful is but a manifestation of the secret
laws of Nature, which, but for this appear
ance, had been for ever concealed from us.
76 The Uses and Triumphs of Mathematics.
' I do not wonder/ says Emerson, ' that
Newton, with an attention habitually engaged
on the paths of planets and suns, should
wonder what the Earl of Pembroke found
to admire in stone dolls.5
Each department of Mathematics has its
own peculiar beauty. The especial beauty
of Geometry consists in its being the most
perfect type of the Deductive Method in
general. In Geometry, too, there is no differ
ence of style ; in a geometrical demonstration
we are unable to distinguish by internal
evidence whether it is Euclid's, Archimedes',
or Apollonius'. In this severe necessity of
form, Geometry is unique.
In Geometry, each link of the chain of
reasoning hangs to the preceding, without
any insecurity in the whole. In Geometry,
we tread every step of the ground ourselves,
at every step feeling ourselves firm, directing
our steps to the required end.
The beauty of Analysis, of the analytical
The Beauty of Mathematics. 77
method, is exactly the opposite. Here we no
longer tread the ground ourselves, — we are
carried along as it were in a railway train,
entering in at one station and coming out
at the other without having any choice in
our progress in the intermediate space. In
geometrical reasoning we reason concerning
things as they are ; in analysis, the contrary
is the case. The analyst represents every
thing — lines, angles, forces, mass, etc. — by
letters of the alphabet. All curves are re
presented by what are known as co-ordinates.
His reasonings are merely operations upon
symbols. He obtains his required results
equally well if he has forgotten, or even
does not know, what he is reasoning about.
This, of course, arises from the perfection
of the analysis, — from the entire generality of
its symbols and its rules. It is not possible,
in any other subject than analytical Mathe
matics, to do this ; that is, to express things
by symbols, once for all, and then go on
78 The Uses and Triumphs of Mathematics.
with our reasonings, forgetting all their
peculiarities. Any attempt to do this (for
such attempts have not been wanting) lead
to the most extravagant and inapplicable
conclusions.
That department of Mathematics known
as the conic sections, has an especial beauty,
from the fact that those curves with which
it is concerned are the curves in which the
planets, comets, etc., move around the Sun,
and in which, also, the moons or satellites
move around their planets.
But as the principal use of the science of
Mathematics lies in its applications to the
Sciences and Arts, so there, as I have said,
also lies its chief beauty. The pleasure
which a temple, or a palace, or a bridge
gives the eye, is that an order and method
has been communicated to stone and iron,
so that they speak and geometrize, becoming
tender or sublime with expression. What in
a great measure gives to Architecture those
The Bea^lty of Mathematics. 79
beautiful curves and angles which we admire
so much ? — what but the labours and dis
coveries of Geometers ? What figure is
more often repeated than that of the circle ?—
that curve which meets the eye often enough
as we go about our daily task. It is brought
before us in the wheels of every vehicle we
meet ; we behold it in the plates and dishes
from which we eat, in the cups and glasses
from which we drink. It is the most beauti
ful, the most perfect, the most useful, and
yet the simplest of all curves or forms.
Under the head of a snake holding its tail
in its mouth, the ancients adopted it as the
emblem of eternity, which has no beginning,
and which has no end. It is, as a writer has
said, — ' The highest emblem in the cipher of
the world. Throughout Nature this primary
figure is repeated without end. We are all
our lifetime reading the copiousness of this
first of forms.' St Augustine defined the
nature of God as a circle whose centre was
8o The Uses and Triumphs of Mathematics.
everywhere, and circumference nowhere. It
has certainly something Divine about it,
being without beginning and without end,
perfect in form, in beauty, and in power.
Surely, then, a science which will unfold its
manifold uses and beauties, is not without
use and beauty too.
Architecture and Geometry have always
been intimately connected, and, indeed, it
is probable that to Geometry Architecture
owes its origin and rise. For there is a
manifest and oftentimes perfect resemblance
between the tombs and temples of the
ancients and the forms and figures of Geo
metry. And, moreover, the principles, at
any rate, of Geometry must have been
known before Architecture was possible.
Mathematics and Music have also from
the earliest times been closely united. For
Euclid, besides his famous ' Elements,' was
also the author of two books entitled ' The
Divisions of the Scale,' and 'An Introduction
The Beauty of Mathematics. Si
to Harmony/ at that time held to be a part
of Mathematics. Pythagoras and Plato were
also writers on Music.
And the eminent mathematicians, Des
cartes, Euler, and D'Alembert, were writers
on the subjects of Harmony and Counter
point. Of them D'Alembert said, — ' It
is solely by closely observing facts, by
reconciling one with the other, and by
making them all, if possible, depend upon
some single fact, or at most upon a very
few principal ones, that they can succeed
in giving to Music a correct, lucid, and un
exceptionable theory.'
The illustrious astronomer Sir William
Herschel was originally a poor organist.
Desiring to study the theory of Music, he
applied himself to the perusal of a treatise
on Harmony. Finding that for a complete
comprehension of the work he required some
knowledge of Mathematics, he applied him
self to this new study, and having mastered
F
82 The Uses and Triumphs of Mathematics.
Geometry and Algebra, the science so fasci
nated him that it came to occupy the first
place in his mind. And he often, after a
fatiguing day's work ®i fourteen or sixteen
hours with pupils, repaired for recreation to
what many would deem these severer exer
cises. He would, I think, hardly have done
this had he not perceived Beauty as well
as Use in the science.
In Painting, too — a subject in which all
educated persons feel a lively interest — where
would those grand effects of distance, of
solidity, etc., be, without Perspective ? — a
branch of Geometry and Optics.
Thus, not with science alone, but also
with the Fine Arts, is Mathematics intim
ately connected. Of the intrinsic beauties
which the science of Mathematics possesses,
I have made mention. But, as I have said,
its chief beauty is owing to its being a some
times the most powerful factor in the dis
covery of a new Law of Physics, and to its
The Beauty of Mathematics. 83
intimate connection with the Fine Arts.
For that which is associated with, and is
an aid to, what is beautiful, cannot fail in
itself to possess beauty too. For whatever
sympathises, is of precisely the same nature
as that with which it sympathises.
It is the office of Art to embellish, to
beautify, wherever or whenever opportunity
offers.
Of Mathematics, therefore, as applied to
and part of the Fine Arts, as well as to
the demonstration of the existence every
where in Nature of fixed, eternal, and
immutable Laws, may it not be said,—
' Beauty chased he everywhere?
'In our lecture-room we teach the letters of the
alphabet ; in our laboratory their use As soon
as these signs, letters, and words have become formed
into an intellectual language, there is no longer any
danger of their being lost, or obliterated from his mind.
With a knowledge of this language he may explore
unknown regions, gather information, and make dis
coveries wherever its signs are current. This language
enables him to understand the manners, customs, and
wants prevailing in those regions. He may, indeed,
without this knowledge, cross the frontiers of the known,
and pass into the unknown territory, but he exposes
himself to innumerable misunderstandings and errors.
He asks for bread and he receives a stone.' — JUSTUS
LIEBIG, ' Letters on Chemistry.'
' It may be laid down as a general rule for Electrical
students, that he who has not a quantitative (i.e. mathe
matical) knowledge of the principles of Electrical Science
will only waste his time in making original experiments.'
— JOHN PERRY, * Electricity in the Service of Man.'
CHAPTER VI.
THE ATTRACTIONS OF MATHEMATICS.
' Nature has made it delightful to man to know, dis
quieting to him to know imperfectly, while anything
remains in his power that can make his knowledge
more accurate or comprehensive.'
Dr THOMAS BROWN.
ONE of the principal attractions of the
Science of Mathematics lies in the fact of
its explaining the why and wherefore of
so much, and in rendering us capable of
reading and understanding ourselves many
great discoveries which we must remain
ignorant of or take for granted.
The celebrated Locke, * who was in
capable of understanding the ' Principia '
from his want of mathematical knowledge,
1 Life of Newton.
88 The Uses and Triumphs of Mathematics.
inquired of Huygens if all the mathema
tical propositions in that work were true.
When he was assured that he might de
pend upon their certainty, he took them
for granted, and carefully examined the
reasonings and corollaries deduced from
them. In this way he acquired a know
ledge of the physical truths of the ' Prin-
cipia,' and became a firm believer in the
discoveries it contained. Of his annoyance
experienced through his want of mathema
tical knowledge, or the pleasure he would
have experienced from a complete perusal
of the ' Principia/ I need make no mention.
The student of geometrical drawing knows
the how of his subject, but (if he be un
acquainted with the elements of geometry)
the why is hidden from him. The practi
cal part of his subject is known to him,
the theoretical is not. But it is by theory
that practical men are rendered of service
to the world. In the science of Mathe-
The Attractions of Mathematics. 89
matics also, no one is able to pass through
a course of Mathematics in a few sen
tences, and but few facilities are given for
acquiring that superficial acquaintance with
facts which enable their possessor to shine
in society, without really enriching his
understanding. In Mathematics, the very
comprehension of the theorems to be de
monstrated and problems to be solved
implies an exercise of the faculties ana
logous to that by means of which every
step of the demonstration is successfully
traced out. And even genius can make
no royal road to learning here. The
greatest genius, perhaps, that ever lived,
viz., Sir Isaac Newton, it is true, assumed
the propositions proved in Euclid's ' Ele
ments of Geometry ' as self-evident truths ;
but in a letter to a friend he regretted
that he had not studied the writings of
o
Euclid with that thoroughness and atten
tion which so excellent a writer deserved,
9O The Uses and Triumphs of Mathematics.
before passing on to the works of Des
cartes and other algebraic writers.
The achievements and triumphs of Mathe
matics are essentially those of patient in
dustry and study, even when treated with
the most masterly skill. This may not seem
an attraction at all to many, in fact, rather
the reverse, but to a man possessed of fair
ability, and of sufficient determination to
enable him to stick to his work, this is a
great advantage. Of course, the brilliant, if
he be a worker, will always surpass the
non-brilliant in whatever branches of Art,
Science, or Business he may happen to be
placed ; but still, I think, brilliancy counts
for less in this subject than in any other.
But the great secret of success in every
thing is not luck but work, and that faculty
known as ' common sense.'
* Common sense/ said Guizot, * is the
genius of humanity.' Common sense is
certainly the genius of Mathematics.
The Attractions of Mathematics. 9 1
1 1 was informed/ said the celebrated
mathematician Stone, ' that there was a
science called Arithmetic. I purchased
a book on Arithmetic, and learnt it. I
was told there was another science called
Geometry. I bought the books, and learnt
Geometry. By reading, I found that there
were good books on these two sciences
in Latin. I bought a dictionary and learnt
Latin. I understood that there were good
books of the same kind in French. I
bought a dictionary and learnt French, this,
my lord, is what I have done. It seems
to me that we may learn everything
when we know the letters of the alphabet/
The Science of Mathematics enables us
to draw correct logical conclusions according
to definite rules ; it teaches us a peculiar
language which, by the aid of signs and
symbols, allows us to express such con
clusions in the simplest manner possible,
intelligent to every one of those \vho
92 The Uses and Triumphs of Mathematics.
understands the language. Before we can
comprehend the results, we must learn the
language, and this is the part which is not
attractive. But having learnt this language,
the reward for our labour is most ample.
We are enabled to become acquainted with
discoveries and truths formerly obscure
and unknown to us, or we may be enabled
to make some original investigations.
For discoveries are not (as we are so
often informed) the result of accident ; the
discovery of the aberration of light by
Bradley was not the result of accident, nor
of the orbits of the planets, nor of Uni
versal Gravitation. ' Malus did not, by
turning round and looking through a prism
of calcareous spar, accidentally discover the
polarisation of light by reflection, but by
considering the position of the prism and
the window ; he repeated the experiment
often, and by virtue of the eminently dis
tinct conceptions of space which he pos-
The Attractions of Mathematics. 93
sessed, he was able to resolve the pheno
menon into its geometrical conditions.' *
Facts (no matter how noticed by the
observer) can only become a part of exact
knowledge when the discoverer's mind be
already provided with precise and suitable
conceptions by means of which he may
analyse and connect them.
The fact that a beam of sunlight on
passing through a prism throws upon the
opposite wall a spectrum of different colours,
has been noticed by hundreds without
their ever inferring that which has helped
to make the name of Sir Isaac Newton
immortal.
Accidents are the theme of the spiritual
ist, not of the arithmetician. The most
casual and extraordinary event — the data
being large enough — is a matter of fixed
calculation. * Everything which pertains to
the human species,' said Quetellet, ' con-
1 See Dr Whewell, 'Phil, of Induct. Sciences,' vol. ii. 190-1.
94 The Uses and Triiimphs of Mathematics.
sidered as a whole, belongs to the order
of physical facts.'
The greatest ' attraction ' of the science
of Mathematics lies, of course, in its appli
cations and uses, and its aid to science ;
and it is solely from a want of mathematical
knowledge that a great number of people
are deterred from the study of many scien
tific subjects. But the days have passed
when the acquirement of knowledge is a
matter of indifference to the general public.
Is it not now rather a matter of universal
emulation ? And the knowledge of Nature
is by no means exhausted. Many great
discoveries yet await the student of Physical
Science and Mathematics. And let it be
remembered that any one who can discover
any one new fact in any science, or assist
others to do so, has thereby rendered the
life of man more glad, and more productive
of benefit and of good to others, than it has
hitherto been in this world of ours.
'Rhyme soars and refines with the growth of the
mind. The boy liked the drum, the people liked an
overpowering Jew's-harp tune. Later they like to
transfer that rhyme to life, and to detect a melody
as prompt and perfect in their daily affairs. . . . By-
and-by, when they apprehend real rhymes, namely, the
correspondence of parts in nature — acid and alkali, body
and mind, man and maid, character and history, action
and reaction — they no longer value rattles and ding-
dongs, or barbaric word-jingle. Astronomy, Botany,
Chemistry, Hydraulics, and the elemental forces have
their own periods and returns, their own grand strains of
harmony not less exact They furnish the poet with
grander pairs and alternations, and will require an equal
expansion in his metres.' — R. W. EMERSON.
CHAPTER VII.
THE POETRY OF MATHEMATICS.
' Poetry is the record of the best and happiest of moments
of the best and happiest of minds.' — SHELLEY.
MATHEMATICS is not without poetry. This
statement may be new to some of my
readers. Of Mathematics as a cure for
mind - wandering ; of Mathematics as the
most perfect type of the deductive method ;
of Mathematics as a great auxiliary to
science, we have not unfrequently heard ;
but of Mathematics as a subject possessing
poetry, I think little indeed has been said.
The poetry which the Science of Mathe
matics possesses may be a poetry quite
its own, but which, I maintain, is poetry
all the same. For what is Poetry ? Not
words, nor yet rhymes, for verse faultless
G
98 The Uses and Triumphs of Mathematics.
in form may be utterly destitute of true
poetry. Poetry is
' No smooth array of phrase.
Artfully sought and ordered though it be,
Which the cold rhymer lays
Upon his page with languid industry.'1 *
' But high and noble matter, such as flies
From brain entranced, and filled with ecstasies? 2
It is not the ear which tells us what is
poetry and what is not, it is our innate
feeling of truth and beauty. If, then, poetic
genius can exist independent and in spite
of phraseology, then many whom we have
not been accustomed to call poets must
be reckoned such ; and much which we
have hitherto regarded not only as not
poetry, but as, perhaps, its very opposite
be so called in the highest sense of the
word.
Euclid's ' Elements of Geometry ' is a
1 Bryant. 2 Emerson.
The Poetry of Mathematics. 99
book of poetry, one of the grandest the
ancients have left us. In the simplicity of
its first principles, the clearness and beauty
of its demonstrations, the wonderful and re
gular concentration of its different parts, and
the universality of its applications, it pos
sesses a power and beauty such as no other
subject can boast of. In most branches
of Art and Science the moderns have far
surpassed the ancients, but, after a lapse
of more than two thousand years, this great
composition of the ancients still maintains
it original pre-eminence and grandeur, and
has acquired additional celebrity from the
fruitless attempts which have been made
to create such another work. Does the
'Principia' of Newton possess no poetry?
It was of this book that the illustrious
H alley said, c So near the gods, man
cannot nearer go;' and Laplace, 'placed it
Note. — To found a superior system of Geometry upon
and by the aid of Euclid is not to create such another work.
ioo The Uses and Triumphs of Mathematics.
above all other productions of the human
intellect/ And the great American philo
sopher Emerson says, ' Newton may be
permitted to call Terence a playbook,
and to wonder at the frivolous taste for
rhymers ; he only predicts, one would say,
a grander poetry ; he only shows that he
is not yet reached — that the poetry which
satisfies more youthful souls is not such
to a mind like his, accustomed to grander
harmonies ; this being a child's whistle
to his ear ; that the music must rise to
a loftier strain, up to Handel, up to Beet
hoven, up to the thorough bass of the sea
shore, up to the largeness of Astronomy.'
The ' Mecanique Celeste' of Laplace, and
the ' Mecanique Analytique ' of Lagrange,
are grand volumes of poetry, for there is
poetry in a mathematical demonstration
when it is the emblem of some great
difficulty solved, or some wonderful result
simply arrived at. There is an ascending
The Poetry of Mathematics. 101
scale of poetry, from the poetry of Words
to the poetry of Actions, and from the
poetry of Actions to the poetry of Actions
again — only not man's but Nature's.
' Presented rightly to the mind ' (says
Professor Tyndall), 'the discoveries and
generalisations of modern science con
stitute a poem more sublime than has
ever yet addressed the human imagina
tion. The natural philosopher of to-day
may dwell amid conceptions which beggar
those of Milton. Look at the integrated
energies of our world, — the stored power
of our coal-fields ; our winds and rivers ;
our fleets, armies, and guns. What are
they ? They are all generated by a por
tion of the sun's energy, which does not
amount to — — of the whole. This is
2,300,000,000
the entire fraction of the sun's force
intercepted by the earth, and we convert
but a small fraction of this fraction into
mechanical energy. Multiplying all our
IO2 The Uses and Triumphs of Mathematics.
powers by millions of millions, we do
not reach the sun's expenditure. And still,
notwithstanding this enormous drain, in
the lapse of human history we are un
able to detect a diminution of his store.
Measured by our largest terrestrial standards,
such a reservoir of power is infinite ; but
it is our privilege to rise above these
standards, and to regard the sun himself
as a speck in infinite extension, — a mere
drop in the universal sea, We analyse
the space in which he is immersed, and
which is the vehicle of his power. We
pass to other systems and other suns, each
pouring forth energy like our own, but still
without infringement of the law, which re
veals immutability in the midst of change,
—which recognises incessant transference
or conversion, but neither final gain nor
loss. This law generalises the aphorism
of Solomon, that there is nothing new under
the sun, by teaching us to detect every-
The Poetry of Mathematics. 103
where, under its infinite variety of appear
ances, the same primeval force. The energy
of Nature is a constant quantity, and the
utmost man can do in the pursuit of
physical truth, or in applications of phy
sical knowledge, is to shift the constituents
of the never-varying total, sacrificing one
if he would produce another. The law of
conservation rigidly excludes both creation
and annihilation. Waves may change to
ripples, and ripples to waves ; magnitude
may be substituted for number, and number
for magnitude ; asteroids may aggregate to
suns, suns may invest their energy in florae
and faunae ; and florae and faunae may
melt in air — the flux of power is eternally
the same. It rolls in music through the
ages, whilst the manifestation of physical
life, as well as the display of physical
phenomena, are but the modulation of its
rhythm.' 1
1 ' Heat a Mode of Motion,' pp. 502-503.
IO4 The Uses and Triumphs of Mathematics.
It has been said ' Science does not know
its debt to Imagination/ but (after the
above passage) who will deny that the
converse also holds — e Imagination does not
know its debt to Science?' It has been
very wisely said that ' the test of the poet
is the power to take the passing day, with
its news, its cares, its fears, as he shares
them, and hold it up to a divine reason,
till he sees it to have a purpose and
beauty, and to be related to astronomy and
history, and the eternal order of the world.'
So it is, or will be, the test of the mathe
matician to take his Geometry and Calculus,
with its uses, its beauties, and its triumphs,
as he shares them, and beholding therein
both Truth and Beauty, show its relation
with every-day life, and bring it down to
the minds and comprehension of the teem
ing millions.
If Mathematics be unpoetical it is false,
and if poetry be illogical it is unreal ; for
The Poetry of Mathematics. 105
the mathematician must not be devoid of
poetic feeling, and the poet must be a true
logician. ' Dante was free imagination, all
wings, yet he wrote like Euclid.' Euclid
had no wings, was all restrictions, yet he
wrote like Dante.
' We think that,' said Macaulay, ' as civil
isation advances poetry almost necessarily
declines.' I think not. I think we shall
have a grander poetry, with mightier strains
of harmony, with loftier modulations, with
mightier rhythms. The greatest of poets
has said :—
' As the imagination bodies forth
The forms of things unknown, the poefs pen
Note. — By mathematician I do not mean a calculating
machine, — that is, a man who, favoured by a good memory,
may have rendered himself intimately acquainted with every
theorem of mathematics, but who is totally unable to propose
a problem for solution. When he possesses the capacity and
talent of proposing a question to himself, and testing the
truth of his calculations by experiment, he becomes qualified
to investigate Nature. For from whence should he derive
his problems if not from Nature ?
io6 The Uses and Triumphs of Mathematics.
Turns them to shapes, and gives to airy nothing
A local habitation and a name?
But the scientific conceptions of to-day
surpass what the most daring of imagina
tions said but yesterday. ' The Comets/
said Kepler (speaking metaphorically), ( are
as numerous in the sky as the fish in the
ocean.' But extending the calculations of
M. Arago, from the planet Neptune to the
furthest limit of the sun's attractive action
we arrive at the appalling minimum number
of 74,000,000,000,000,000 of comets, that
for one of their periods at least are subject
to the empire of the sun.1
I close this chapter with an anecdote
concerning the celebrated astronomer and
mathematician Euler, as related by Arago
to the Chambre des Deputes, at a meeting
on the 23d March 1837. I quote in full,
illustrating so well as it does this portion of
1 ' Les Cometes,' pp. 120-122.
The Poetry of Mathematics. 107
my subject ; it is an anecdote deserving to
be far more widely known than it is.
' Euler, the great Euler, was very pious ;
one of his friends, a minister of one of
the Berlin churches, came to him one day
and said, " Religion is lost ; faith has no
longer any basis ; the heart is no longer
moved, even by the sight of beauties, and
the wonders of Creation. Can you believe
it ? I have represented this Creation as
everything that is beautiful, poetical, and
wonderful ; I have quoted ancient philo
sophers, and the Bible itself: half the audi
ence did not listen to me, the other half
went to sleep or left the church." " Make
the experiment which truth points out to
you," replied Euler. " Instead of giving the
description of the world from the Greek
philosophers or the Bible, take the astrono
mical world, unveil the world such as
astronomical (i.e., physical and mathe
matical) research constitute it. In the
io8 The Uses and Triumphs of Mathematics.
sermon which has been so little attended
to, you have probably, according to Anaxa-
goras, made the sun equal to Peloponnesus.
Very well ! Say to your audience that,
according to exact, incontestable (mathe
matical) measurements, our sun is 1,200,000
times larger than the earth. You have,
doubtless, spoken of the fixed crystal
heavens; say that they do not exist, — that
comets break through them. In your ex
planation, planets were only distinguished
from stars by movement ; tell them they
are worlds, — that Jupiter is 1400 times larger
than the earth, and Saturn 900 times so ;
describe the wonders of the ring ; speak of
the multiple moons of these distant worlds.
Arriving at the stars, their distances, do
not state miles — the numbers will be too
great, they will not appreciate them ; take
as a scale the velocity of light ; say that it
travels about 186,000 miles per second ;
afterwards add there is no star whose light
The Poetry of Mathematics. 109
reaches us under three years, — that there are
some of them with respect to which no
special means of observation has been used,
and whose light does not reach us under
thirty years. On passing from certain re
sults to those which have only a great
probability, show that, according to all
appearance, certain stars would be visible
several of millions of years after having
been destroyed, for the light emitted by them
takes many millions of years to traverse the
space which separates them from the earth."
' This advice was followed ; instead of the
world of fable, the minister preached the
world of science. Euler awaited the coming
of his friend after the sermon with im
patience. He arrived despondent, gloomy,
and in a manner appearing to indicate de
spair. The geometer, very much astonished,
cried out, " What has happened ? " " Ah,
Monsieur Euler," replied the minister, " I
am very unhappy : they have forgotten the
1 1 o The Uses and Triumphs of Mathematics.
respect which they owed to the sacred
temple, they have applauded me" ''
Of a truth there is a thousand times more
poetry in the reality than in the fable : —
' For the world was built in order,
And the atoms march in tune ;
Rhyme, the pipe, and Time, the warder,
Cannot forget the sun and moon.'' 2
1 See ' Les Merveilles Celestes.3 2 Emerson.
' If but one hero knew it
The world would blush inflame;
The sage, till he hit the secret.
Would hang his head for shame.
But our brothers have not read it,
Not one has found the key ;
And henceforth we are comforted, —
We are but such as they.'—R. W. EMERSON.
'Among all men, though all men be unfit (to pene
trate within the temple wherein the Divine Mystery is
enshrined), none can nearer attain fitness to approach
the temple than those who contemplate the mysteries of
Infinite Time and Infinite Space, of Infinite Might and
Infinite Life, all ruled by Infinite and Eternal Law.
They alone perceive what marvels of knowable truth lie
within the infinite domain of the Unknowable.'
Knowledge, ' Science and Religion.'
CHAPTER VIII.
METAPHYSICAL OR SPIRITUALISTIC
MATHEMATICS.
* Never be deceived by words. Always try to penetrate
to realities: — W. J. Fox.
' No difficulty is unsurmountable if words be allowed to
pass without meaning? — LORD KAMES.
IN writing about the uses of Mathematics,
I made some mention of the difference be
tween the ancient and modern ideas on
that part of my subject, — how some of
the ancients valued it, not for its practical
or applied uses, but only as habituating the
mind to the contemplation of truth, and
raising man above the material universe,
and in leading him to the knowledge
of the essential, the eternal, the abstract
truth, — as disciplining the mind (not in the
H
ii4 The Uses and Triumphs of Mathematics.
sense now usually understood), and not as
ministering to the base wants of the body.
But there are some now-a-days who, think
ing they are creating a new era of thought
(when in reality they are only going back
some two or three thousand years ; it is
the old spirit under a new form), have re
sumed these old supposed uses of the
science, and have been good enough to
bestow on us, amongst other things, a
1 Fourth Dimension.'
Of their manner of doing this I give here
two instances, more being superfluous.
The general character of their attempts
is by juggling with the symbols of the
pangeometers, — by appealing to the metageo
— metrical vagaries of Lobatschewsky, Rie-
man, etc. — and by using very long words,
and making up very learned - looking
sentences, which are just as sensible very
often read backwards as forwards. Why
will the public be so taken in by words I
Metaphysical Mathematics. 115
words ! words ! and sentences which they
(or anyone else, as far as that goes) are
totally unable to understand ? one of these
fourth dimensionists actually assuming the
identity of an algebraic multiple with
a spatial magnitude !
Mathematicians employ algebraic quanti
ties of the first, second, and third degree
to denote geometrical magnitudes of one,
two, and three dimensions respectively,
and the fourth dimensionists say (or must
say, if they consistently adhere to their
principles) therefore there must be a
geometrical magnitude of a fourth dimen
sion corresponding to an algebraic quantity
of the fourth degree. But if this be so,
there must be by analogy or common
sense a geometrical magnitude of the
fourth, fifth, sixth, . . . nth dimension
corresponding to an algebraic quantity
of the fourth, fifth, sixth, ... nth degree,
where n may be any number, ten or
1 1 6 The Uses and Triumphs of Mathematics.
ten millions, or ten thousand billion millions,
or any other number you like to write.
For this is really all the evidence we
possess of the fourth dimension, holding,
you see, thus just as good for fifth, sixth,
or sixth billionth millionth dimensional
space ; which is indeed a land of mist
and shadows, a bourne from which no
traveller hath, or is ever likely, to return.
This is, nevertheless, one of the means
whereby some would now attempt to
' demonstrate the existence of another
world.' If it did really require such bol
stering up as this, it would indeed be in
a perilous state.
Another way is as follows : — It requires
a certain amount of conceivability. But
no matter. — It supposes you to imagine ' a
direction which is at one and the same
time perpendicular to what we know as
height, breadth, and length, — that is, per
pendicular to the sides of a box, and yet
Metaphysical Mathematics. 1 1 7
only in one direction. If it be possible
to realise this, further illustration is value
less ; geometrical four dimensional space
is already understood, but, if not, further
illustration is useless.' Quite so. If any
of my readers possess the requisite imagina
tion, he or she then understands four
dimensional space. For my own part, I
can only say / do not. But with regard
to this I just wish to point out one
thing :
If it be possible to conceive a line or
direction at one and the same time per
pendicular to the sides of a rectangular
box, and yet only in one direction, it is
just as possible that ' two straight lines
should enclose a space/ — that ' the whole
should be less than its part,' etc., etc. ;
the whole of Euclid falls to the ground,
Science becomes of no value, and Chaos is
once more triumphant in the world, — a
world maybe —
1 1 8 The Uses and Triumphs of Mathematics.
' Where nothing is, and all things seem,
And we are shadows of a dream?
Why have I inserted the above, does
the reader ask ? NOT with any intention
of being funny, nor yet as so much padding
in order to fill up a certain number of
pages, but because this is an age of em
piricism ; they flourish like the green bay
tree, about which we have heard so much.
But their days are numbered, for nothing
can endure but what is genuine. They
may be — nay, many we know are — self-de-
lusionists, but the misfortune is that they
not only delude themselves, but delude or
attempt to delude hundreds or maybe
thousands of others. Metaphysical Mathe
matics has always been a subject attended
with danger and difficulty, and loss of
both time and labour. Beware of those
who inform you that you must neglect,
must get beyond the vulgar uses of Mathe
matics, and attain to a science which is
Metaphysical Mathematics. 1 1 9
as independent of the actual subject con
sidered as geometrical truth is independent
of an ill-drawn diagram. Had Euclid, in
stead of explaining and demonstrating the
properties of lines and curves, called upon
men to reverence the mystery of Mathema
tics, we should have had a creed of Geo
metry, in the place of a Science, and our
architects believing in tunnels and bridges
instead of building them.
Metaphysics is a science too often re
sembling the ox of Prometheus, a sleek,
well-shaped hide, stuffed with rubbish,
goodly to look at, but containing nothing
to eat. And does Philosophy possess that
supreme pre-eminence generally assigned
it ? By its study we obtain a knowledge
of the intellectual world, the laws of thought,
of mental inquiry, and of the spiritual
nature of man ; but, nevertheless, philo
sophy has not been able to prevent people
from being burnt for witchcraft, for when
i2o The Uses and Triumphs of Mathematics.
the illustrious Kepler went to Tubingen
to save his mother from the stake, he
succeeded only by proving that she pos
sessed none of the characteristic signs
essential to a witch. And the discovery
by Descartes of algebraic geometry is, to
my mind, a greater discovery than his
cogito ergo sum (I think, therefore I am),
and of more beneficial use to mankind at
large. But then there is philosphy and
philosophy.
But to return from this digression. The
ancient philosopher Socrates, when speak
ing of the ancient metaphysical speculators,
etc., of his day, demanded of such inquirers
whether they had attained a perfect know
ledge of human things since they searched
into heavenly things, ' or if they could think
themselves wise in neglecting that which
concerned them, to employ themselves in
that which was above their capacity to
^lnderstand?
Metaphysical Mathematics. 121
The spiritualists of to-day would do well
to bear these words in mind, for they are
full of wisdom ; and wiser, perhaps, are
the words of Emerson : ' Let us know
what we know for certain. What we have,
let it be solid, seasonable, and our own.
A world in the hand is worth two in the
bush. Let us have to do with real men
and women, and not with skipping ghosts.'
There are problems in Metaphysics and
Mathematics which may be demonstrated
to be insolvable. To describe the limit
of the human power with respect to these
problems is not yet possible. Neverthe
less the capacities of our understanding
will probably one day be well considered,
and the line drawn between what is and
what is not comprehensible by us.
That such a thing as some Psychic Force
exists, whose laws scientists are at present
unable to explain, is both possible and con
ceivable. But we must remember that
122 The Uses and Triumphs of Mathematics.
Attraction or Gravity 1 was known to
exist hundreds, nay thousands, of years
before mankind was able to discover the
law that it obeyed, so we should await
the coming of that second Newton to ex
plain this Psychic Force (if it exist), and
not assign a supernatural cause to what
science is certain in time to explain.
And be not deceived by the wonders
that the spiritualists say man is able to
perform, aided by this Psychic Force.
These marvels are paltry as compared
with those which man has been able to
achieve by the assistance of Science. Of
some of the wonders you have read in
Chap. III., and there are others no less
wonderful. — Man, aided by Science ', can tell
1 ' Gravity is often incorrectly spoken of as being a Force.
Gravity is a name for the general fact that any two material
bodies, if free to move, approach each other with a gradually
increasing swiftness ; Force is the name which we give to
the unknown cause of this fact.'— PROFESSOR HUXLEY.
It is now possible (by means of the spectroscope) to detect
the presence of T 8 o o'o o ootn Part °f a grain of salt.
Metaphysical Mathematics. 123
you the exact number of waves of light
emitted from the sun per second, and their
exact length ; he can tell you what a star,
billions of miles out there in space, is made
of; he has surpassed the old miracles of
Mythology, flying across the sea, and send
ing his messages under it ; the artist (aided
by Science) can display to your astonished
gaze true and realistic pictures of what this
earth was like ten thousand or ten hun
dred thousand or ten millions of years ago.
Man is to-day able to speak, and his de
scendants, whose grandparents are yet un
born, shall hear his voice. The man of
science is able to make a jet of gas twenty
feet distant from him sing, and to continue
its song for hours, loud enough to be heard
by an assembly of a thousand people. The
comparative anatomist, possessing but a
small fragment of a bone, a tooth, is able to
relate the whole history of this being be
longing to a past world, describe its size
124 The Uses and Triumphs of Mathematics.
and shape, point out the medium in which
it lived and breathed, and demonstrate
whether its nourishment consisted of animal
or vegetable food, and its organs of motion.
And the chemist is able, knowing the
proportion in which any single substance
unites with another substance, to assign
the exact proportion in which the former
will unite with all other bodies whatever.
Such are a few of the marvels of
Science.
From men who resort to pan-geometry
and logomachy to prove the existence of
another world, what may not be expected ?
what can we hope ? The man of Science
has achieved these triumphs, and may safely
assert all of them as realities, because he
has acquired a knowledge of natural pheno
mena, and an intimate acquaintance with
natural laws, and because everything being
subject to definite Laws, when these Laws
are known, the rest follows from them.
Metaphysical Mathematics. 1 2 5
Many of these triumphs we are, of course,
able to verify for ourselves.
The difference between the absolute know
ledge of the man of Science and the em
piric or spiritualist is well illustrated in that
anecdote of Socrates : ' Men call me wise.
Certainly I know little ; I will inquire.' He
then questioned many people, and indeed
found that they knew little, but that they
thought that little much. ' In fact,' said
Socrates, at length, ' though I know as
little, yet in one sense I am their superior ;
I know how little that little is, whereas
they are ignorant how ignorant they are.'
It is quite true that the man of Science
(in one sense) does not know
' How the chemic atoms play,
Pole to pole, and what they say.'
EMERSON.
Nor
' . . . What wove yon woodbirds vest,
Of leaves and feathers from her breast ?
126 The Uses and Triumphs of Mathematics.
Or how the fish outbuilt her shell,
Painting until morn each annual cell?
Or how the sacred pine-tree adds
To her old leaves new myriads V
EMERSON.
Nor does the spiritualist, though he pre
tends to have solved the mystery of life.
He knows no more than his fellows, only
less, because his fellows know that they
do not know ; they also know that he does
not know, whereas the spiritualist does not
know that he does not know. We owe all
our knowledge of Nature to Science, we
owe nothing at all directly * to Spiritualism.
The spiritualist allows effects to govern
his will, whilst by a true insight into their
hidden connections he might Govern them.
o o
Having thus warned my readers against
Fourth Dimensionists, Metaphysicians, and
1 During the Middle Ages, and amongst the ancients, the
noble studies of Astronomy, Chemistry, etc., were often cul
tivated as subsidiary to those of Astrology, Alchemy, etc. ;
Alchemy chiefly from 13th to I7th century.
Metaphysical Mathematics. 127
Spiritualists, I will close this chapter with a
passage, and a few remarks on it, taken from
S. Bailey's Essay on ' The Progress of Cul
ture.' In it he says, ' It is unwise for any
one to enter very minutely into the history
of the science to which he devotes himself—
more especially at the outset. Let him per
fectly master the present state of the science,
and he will be prepared to push it further
while the vigour of his mind remains un
broken ; but if he previously attempts to
embrace all that has been written on the
subject, — to make himself acquainted with all
its exploded methods and obsolete doctrines,
his mind will probably be too much en
tangled in their intricacies to make any
original efforts ; too wearied with tracing
past achievements to carry the science to a
further degree of excellence. When a man
has to take a leap, he is materially assisted
by stepping backwards a few paces and
giving his body an impulse by a short run
1 2 8 The Uses and Triumphs of Mathematics.
to the starting place ; but if his precursory
range is too extensive, he exhausts his forces
before he comes to the principal effort.7 But
the whole question rests on those two words
—very minutely. For, in order that we may
share in what men are doing in the world,
we must share in what they have done.
' For to know certain general symbolical
results — that is to say, certain modern ana
lytical methods, which are supposed to
render all scientific history superfluous — is
an accomplishment which can only be of
little value in education ; for a good edu
cation must connect us with the past as
well as with the present, even if such mere
generalities did supply the best mode of
dealing with all future problems, which, in
fact, they are very far from doing/ 1
1 See 'A Liberal Education,' by Dr Whewell.
The Future hides in it
Gladness and sorrow :
We press still thorow :
Nouglit that abides in it
Daunting us — Onward .'
And solemn before us,
Veiled the dark Portal,
Goal of all Mortal.
Stars silent rest o'er us —
Graves under us, silent.
While earnest thou gazest,
Comes boding of terror,
Come phantasm and error :
Perplexes the bravest
With doubt and misgiving.
But heard are the voices,
Heard are the Sages,
The Worlds and the Ages :
" Choose well : your choice is
Brief, and yet endless"
Here eyes do regard you
Iti Eternity's stillness ;
Here is all fulness,
Ye brave to reward you ,
Work, and despair not? — GOETHE.
First quoted by T. Carlyle in ' Past and Present.7
I
CHAPTER IX.
CONCLUSION.
'Any intelligent man may now, by resolutely apply
ing himself for a few years to mathematics, learn more
than the great Newton knew after half a century of
study and meditation.' — LORD MACAULAY.
IF some of the statements made in Chap
ter III. should seem to be too wonderful to
be credited, or if the nature and difficulties
of some of the problems which the science
of Mathematics has so successfully solved
appear to overwhelm the mind, let it be
remembered that the science of Mathematics
has ever lived and never dies. One mathe
matician dies, but his works remain ; another
takes up the work where he left it, and the
1 3 2 The Uses and Triumphs of Mathematics.
chain of reasoning is unbroken, the work
carried on.
Commencing his calculations thousands of
years ago amongst some of the nations of
the East, in Babylon he toiled, and amongst
the Egyptians he found a dwelling-place.
Among the temples of India, the pagodas
of China, the pyramids of Egypt, and the
plains of Arabia he thought and studied.
When Science fled to Greece, his refuge
was in the schools of her philosophers ; and
when darkness and bigotry covered the face
of Europe for hundreds of years, he pursued
his studies amidst the burning plains of
Arabia. When Science returned again to
Europe, the Mathematician was there, toiling
in Leonardo Bonacci, suffering in Galileo,
triumphing in Descartes, and triumphing still
more in Leibnitz, and Newton, justly regarded
as the greatest genius that ever lived.1
1 And soon after in Laplace (born 1749), and Lagrange
(born 1736), and Euler (1707), etc.
Conclusion. 133
Standing on the lofty pinnacle of the
Temple of Science of the present day, of
which we are so justly proud, and which
Mathematics has so powerfully and so effec
tually helped to raise, and looking around us,
we become aware of the deep debt which
the world owes to original discoverers in
this Science, and we see what an import
ant, though not openly apparent, part it has
played in the history of the world.
What mathematicians will accomplish in
the future remains to be seen, but one thing
we know, the past and the present constitute
one unbroken chain of reason, condensing all
time to the mathematician into qne mighty
NOW. We shall not live to behold these
anticipated triumphs of mind over matter,
but who can doubt the final result ? Look
back to the ancient mathematicians ; com
pare their power and knowledge with those
of the modern, grasping in a few years of
patient study far more than his predecessor
134 TJie Uses and Triumphs of Mathematics.
was able to learn in a lifetime. Are the
problems remaining to be solved more dif
ficult, more inaccessible, than those which
have been so successfully solved ?
The results recorded by the ancient Ma
thematicians are of inestimable value in the
solution of some of the most difficult pro
blems of to-day ; similarly the records made
now shall descend to generations yet unborn,
and aid them in the same manner as the
records made thousands of years ago aid the
mathematicians of the present day.
In conclusion, then, Science (in its broad
est sense) is the great power of the day, and
it is, as W. P. Fox says in his Lectures to
the Working Classes, the friend of man ; the
history of its advance is the history of human
progress ; it sheds a light on the past, and
by doing so, in some measure illuminates the
coming future ; it is in harmony with the
being and well-being of all the inhabitants of
this world of ours ; and in proportion as it
Conclusion. 135
makes known to us the great principles and
influences that pervade Creation, it makes
us at one with Creation, and the recipients
of its good and its blessings. Science is
the friend of man, raising and dignifying
man, and qualifying him more and more for
the full possession of his rights, the exercise
of his powers, and the accomplishment of
whatever is good and great in this world,
and of all that its various means and appli
ances are capable of rendering.'
Mathematics is one of — sometimes the
greatest — auxiliary to Science ; by Science
are the inner works of Nature reverently
uncovered. I commend, therefore, the study
of Mathematics to you as worthy of all your
acceptation, only bidding you remember
that
' Industry is the -VWE. philosopher* s stone]
and that any one who is able to decipher
any of the hieroglyphics of the volume of
136 The Uses and Triumphs of Mathematics.
Nature, or to carry any Science to a further
degree of excellence, has not lived in vain,
but has added something to the sum of
human happiness and human knowledge.
FINIS.
APPENDIX.
APPENDIX.
THE SQUARING OF THE CIRCLE.1
THERE are four famous problems which from
time immemorial almost have had a multitude of
patient devotees. They are : — The discovery of
perpetual motion ; the trisection of any angle ;
the finding of two mean proportionals between
two given straight lines (often referred to as the
duplication of the cube) ; and the quadrature or
' Squaring ' of the Circle.
With regard to the first, I make no further
mention here than to suggest that before any one
attempt its solution, he should read, mark, learn,
and inwardly digest the principle of the conserva
tion of energy, and he will then comprehend the
absurdity of his attempt.
With regard to the second and third — The Tri-
1 See 'Budget of Paradoxes,' by Professor De Morgan, being
a series of papers in the A'hencciun for 1863, and subsequent years.
1 40 Appendix.
section of any Angle, and the Duplication of the
Cube, they are not unsolvable or impossible pro
blems, but only so by means of elementary geo
metry, for by the postulates of ordinary geometry
all constructions must be made by the aid of a circle
and undivided ruler. Now straight lines intersect
each other only in one point, and a straight line
and a circle intersect each other only in two points.
But the trisection of any angle, or the duplication
of the cube, requires for its solution the inter
section of a straight line and a curve, — of what is
known as the third degree, or two conies ; but
all of these are excluded by the postulates of ordin
ary geometry. If the postulates of elementary
geometry allowed that a parabola or an ellipse
could be described with what is known as a
given focus and directrix, as they allow that a
circle can be described with a given centre and
radius, then these two problems are solvable by
elementary geometry, their so-called insolvability
being merely a restriction placed (by whom un
known, but prior to Euclid) upon the postulates
of ordinary geometry.
Passing on now to the Quadrature, more gener
ally known as the Squaring of the Circle, the ques
tion which arises is, What is the Squaring of the
Circle ? It is to make a circle containing exactly
Appendix. 1 4 1
the same area as a square. And the solution of
this problem depends on finding the precise or
exact ratio which exists between the diameter
and the circumference. This may appear at first
sight absurd, for what ratio can possibly exist
between two things so perfectly unlike ? For the
diameter of a circle is a straight line passing
through the centre and terminated both ways by
the circumference, while the circumference is, of
course, a curved line. But supposing the circum
ference of the circle stretched out into a straight
line to its full extent, similarly as a wire ring
might be done, by cutting it through in one point,
and then stretching it out into a straight piece
of wire, what then is the proportion between the
diameter and the circumference ? Many persons
think that it is an easy matter to determine this
ratio — namely, by measuring. First measure the
diameter, and then the stretched out circum
ference, and you have the required proportions,
which, supposing the diameter to be 7 inches,
you will probably find the circumference to be
perhaps 22 inches. This must be right, you say,
for we have measured it. Only it is not: the pro
portion is erroneous. The ratio of the diameter
to the circumference is not exactly as 7 to 22.
And this failure results simply from the nature
142 Appendix.
of the thing. For it is impossible to compare the
physical with the ideal, — the mechanical opera
tions of the finest arts with the pure and simple
abstractions of the mind. For even as in the
Fine Arts there is an ideal beauty which no
artist or connoisseur can or could ever attain, so
in mathematics there is an accuracy of propor
tion, or ratio, which the finest instrument con
structed or ever likely to be constructed by man
can never attain. The above proportion, 7 to 22,
is a very rough approximation ; a nearer approxi
mation is possible by measurement, but a rough
approximation is only possible by instrumental
means. Measurement and arithmetic this ratio
surpasses, but ideas and mathematical expressions
are able to reach it. By means of ordinary num
bers we can only approximate to this ratio or
proportion, for this proportion is simply what is
known as an infinite series of which the law can
not be stated in ordinary terms of the decimal
notation. Archimedes found this ratio to be be
tween 3fg and 3-^-°-. By which he meant that the
ratio of the circumference to the diameter lies
between 3-fg- and 3-^ J. And this is approximately
as 22 to 7. But a small error in the circumference
leads to a greater error in the surface or area of
the circle, and to a still greater error in the solid or
Appendix. 143
cubic contents of the sphere ; for errors increase
by multiplication.
Metius, a Dutch mathematician of the seven
teenth century, found the ratio to be 355:113.
The Hindoos, however, had obtained an expres
sion nearly as accurate. From the time of Metius
down to the present day, closer and closer approxi
mation by different mathematical methods have
been obtained.
Ludolph Von Coulen, by simple arithmetical
processes, showed that this ratio was between
3.14159265358979323846264338327950288
and,
3.14159265358979323846264338327950289
a result so accurate, that, says Montucla, 'if
there be supposed a circle whose radius is the
distance of the nearest fixed star (250,000 times
the earth's distance from the sun) the error in
calculating its circumference is so excessively
small a fraction of the diameter of a human hair
as to be utterly invisible not only merely under
the most powerful microscope yet made, but under
any which future generations may be able to
construct ; ' the above result being true to 36
significant figures, a result which will perhaps
demonstrate to the general reader that the ap-
1 44 Appendix.
proximations of mathematics are hardly approxi
mations in the ordinary sense of the word. In
the year 1688 James Gregory gave a demonstra
tion of the impossibility of effecting exactly the
quadrature of the circle, now generally accepted.
It can be expressed in the form of a Definite
Integral, which it is perhaps possible may be
expressed in finite terms containing irrational
numbers, this being hailed, perhaps, as a solu
tion of the grand problem. But be this so or not,
the reader will at once see that the so-called ap
proximations mentioned above are far more than
sufficient for any practical applications ever re
quired by man, even in those most delicate of all
delicate calculations, the calculations of Astronomy.
The solution of the problem mentioned just above,
it is almost needless to say, is not the one attempted
by the ' Squarers,' who, I am sorry to hear, grow
more numerous every day. I can only suggest to
them that when they have arrived at the conclu
sion that 3.1415 or 3.14159, etc., is the exact ratio,
they should carefully bear in mind the result of
Ludolph Von Coulen, stated above, and also medit
ate upon the fact that by other methods its value
is now known exact to 600 places of decimals.
V.T.S. : D. :5-89.
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